Why does Titan have an atmosphere denser than that of the Earth? And why does Europa not have even a fraction of Titans atmosphere given that it is not that small as compared to Titan?
Following is a block quote from How to Get an Atmosphere by Peter Tyson
Saturn's moon Titan belongs to a very select club within the solar system. It is one of only four "terrestrial" planets or moons—those with solid bodies, as opposed to those made largely of gas, like Jupiter and Saturn—that has a substantial atmosphere. The other three that wear blankets of gas are Venus, Mars, and our own Earth.
Why just these four? Why not also, say, Mercury, Jupiter's biggest moons, our moon? How did those lucky four come by their atmospheres?
It turns out that getting an atmosphere—and holding on to it—really comes down to how big and how close to the sun you are (or, for Titan, how close you are to a really big planet). For astrophysicists, it's infinitely more complex than that, but if you just want the quick and dirty answer, that's it, and here's why: Original gas
The story of planetary atmospheres begins back at the beginning of our solar system, when the planets were forming. During that period, the so-called inner planets—Mercury, Venus, Earth, and Mars—all developed the same kind of air, a so-called primary atmosphere. It consisted mostly of hydrogen and helium, the two elements that today make up 98 percent of the sun and gas giants like Jupiter.
Like planet-sized magnets, the proto-planets had sufficient gravity to draw these two gaseous elements in from the solar nebula, the vast cloud of gas and dust that surrounded the sun early in the solar system's history. In that primordial time, the sun was not very bright and thus not very hot, and this allowed the four inner planets to hold onto those atmospheres.
Three factors play into a gas's ability to escape the pull of a planet's gravity: temperature, molecular mass, and escape velocity (the speed a molecule needs to achieve to escape into space). Hotter, lighter, and faster particles more easily slip out of a planet's gravitational grip into space than cooler, heavier, and slower particles.
Hydrogen and helium are two of the lightest molecular-weight molecules out there. And as the sun grew brighter and hotter, the molecules of hydrogen and helium that the four inner planets had been able to retain became hotter and faster, finally reaching escape velocity. When that happened, perhaps within a few hundred million years after the formation of the inner planets, these gases escaped into space, leaving Earth and its three companions little more than balls of rock in space.
The four giant outer planets, meanwhile—Jupiter, Saturn, Uranus, and Neptune—were able to keep their hydrogen and helium because of their size: their gravitational pull is mighty enough to contain those two light gases, and the sun is too far away for its heat to make any difference. So those four gas giants still host their primary atmospheres. Putting on air
Fortunately for us, there are secondary atmospheres, otherwise we wouldn't be here. These are atmospheres that arise long after a planet's primary atmosphere has vanished into the ether. Yet not all rocky bodies have the means to sustain them. (Mercury, for one, is too close to the sun to hold onto any type of gas.) How did the four solid bodies that have them win the atmospheric lottery?
Leaving Titan aside for the moment, Earth, Mars, and Venus all began developing their secondary atmospheres in the same way. Over time their envelopes of air would become as unlike as heaven and hell—in the case of Earth and Venus, for example—but initially they likely appeared largely the same. The reason is that, despite their differences today, these three planets lie in roughly the same neighborhood of the solar system and are thought to consist of roughly the same mix of elementary stuff.
Earth became heavenly, Mars froze solid, and all hell broke loose on Venus. What happened?
While Earth, Mars, and Venus eventually got to the point where they could no longer embrace hydrogen and helium, they did have sufficient gravity and cool enough surface temperatures to retain heavier molecular-weight gases like carbon dioxide and water vapor. And they had plenty of these two substances stored away in one form or another within their stony bodies. The CO2 and H2O came from two sources: the original building blocks out of which the planets formed as well as comets that regularly slammed into the planets early in their history.
Fortunately, again, for us, these crucial substances of CO2 and H2O—and also nitrogen, which comprises 78 percent of our atmosphere—were not irretrievably locked in the rocks. These substances had a catalyst that helped free them: heat. Within each planet, a molten core created during the planet's initial formation released heat, and so did the slow decay of radioactive elements deep beneath the surface. This heat kept each planet toasty enough to produce volcanic eruptions, which spewed these gases out of the interior.
Despite increased warmth from the sun, these heavier molecules could not escape the gravity of Earth, Mars, and Venus, respectively, and so they began building up just above each planet's surface. The result was a secondary atmosphere—or what most of us know simply as the air.
But, in time, Earth became heavenly, Mars froze solid, and all hell broke loose on Venus. What happened? From heaven to hell
This is where the how-close-you-are-to-the-sun part comes in. On Earth, all that water vapor belched out of volcanoes condensed in the young atmosphere into liquid water, then fell to the surface as rain. Over eons, this formed the oceans. Most of the CO2, meanwhile, became incorporated into the seas and into sedimentary rocks. Most, but not all, and this is crucial: enough CO2 remained as gas in the atmosphere to create the greenhouse effect that keeps our planet a life-sustaining average global temperature of about 59°F. Everything eased into a wonderful balance—all brought about by our ideal distance from the sun.
As for Mars, its secondary atmosphere had two strikes against it from the start: the planet's size (too small) and its distance from the sun (too far). In its first 500 million years or so, the Red Planet had a warm atmosphere and liquid-water oceans, just like Earth. But Mars is so small that its internal heat engine burned out early on, and it is so far away from the sun that all the water vapor that its once-active volcanoes had erupted eventually froze out of the atmosphere, becoming trapped beneath the surface as ice. All this left the Red Planet as cold and barren and apparently lifeless as the moon. Mars still has an atmosphere, but its pressure is 100 times less than Earth's and it's almost entirely composed of CO2—about the last thing we'd want to breathe.
Venus has roughly the same concentration of CO2 as Mars, yet its atmosphere went in precisely the opposite direction. Size wasn't an issue: Venus has about the same mass as Earth so is plenty hot within. But distance from the sun has made all the difference. Venus is near enough to our star that all the water vapor released from its volcanoes burned off long ago, and without liquid water, the planet could not form oceans that could absorb the CO2.
The result has been a runaway greenhouse effect. While a greenhouse effect raises the temperature of Mars by about 5°F and Earth by about 35°F, on Venus it has jacked up the temperature by around 500°F. The resulting atmosphere is truly nasty from our perspective: hotter than a self-cleaning oven, with a density about 10 percent that of water and a pressure about what you'd feel a half mile down in the ocean. A moon with atmosphere
And what about Titan? Why did it get an atmosphere when, for example, none of Jupiter's big moons, which are a lot closer to the sun, did? Well, in this case, distance from the sun doesn't really come into it; the moons of the outer planets are so far away that it's a moot point. But distance does factor in—distance to a giant planet. And, again, size matters. In fact, a moon needs the right balance of nearness to a giant neighbor and adequate gravity—that is, size—to gain and hold an atmosphere, and of all the moons in the solar system, only on Titan did Nature strike that balance.
Whether we humans could ever severely or permanently alter our own atmosphere is unknown, but do we really want to take that chance?
Titan is close enough to Saturn that it gets squeezed by tidal forces powerful enough to heat up its interior. So the volcanic activity that long ago died out, for instance, on our similarly sized moon, has continued there. That activity releases CO2 and water vapor, but since Titan's mean surface temperature is -289°F, both of those quickly fall out as ice on the surface. That leaves nitrogen, which remains a gas at that temperature, and methane, which builds up in an interaction between sunlight and CO2 ice. The result is an atmosphere that is roughly 90 percent nitrogen and 7 percent methane. (Interestingly, as radically different as Titan's atmosphere is to our own, it is still worlds closer in composition and pressure to Earth's nitrogen-rich air than are the CO2-dominant atmospheres of either Mars or Venus.)
Saturn makes Titan's gases come out; Titan's size ensures some of them stick around in an atmosphere. Jupiter's moon Io, being so close to its humungous neighbor, has plenty of volcanic activity, but the moon's mass is too small to wield the kind of gravity needed to maintain a hold on the gases that gush out of its insides.
While Europa (and Ganymede and Callisto) are indeed not too different in size from Titan, they are too close to the Sun for anything more than a thin atmosphere to form.
Jupiter and its moons are far away enough from the Sun to allow for water to condensate and thus form icy moons. At ~5 A.U. there is however still too much sunlight for other volatiles, like the ones found in Titan's atmosphere at 9 to 10 AU to condense.
Titan being further away and thus colder, allowed a surface consisting of mostly water ice and ammonia ice. The ammonia ice reacted over time through photolysis or possibly from internal heat and N2 gas was released, forming an atmosphere.
This book on the bodies in our solar system that have a (subsurface) ocean has a chapter on Titan and its atmosphere.
A final determining factor for the presence of an atmosphere is something you allude to in your question, which is the size, or more precisely, the mass of the moon. You of course need a certain amount of gravity in order to keep all the gases that would form your potential atmosphere from flying off into space.