Bulk information regarding anything in this world is available in a single touch through Internet. My question is whether it is possible to replace this Internet with a constellation of Cubesats in low Earth orbit for transferring data all over the world more securely than over Internet?
You can't replace the Internet like that since Internet is also a collection of protocols for routing and addressing, and transmitting users' data. You can try swapping fiber optic links for LEO sat comm ones, but throughput will invariably suffer and connection will be less reliable.
Satellites' transponders are the bottleneck - instead of an hierarchy of ISPs and various telecom companies with widening link capacities as you go up the "food chain" you have to put maximum imaginable downlink/uplink capacity into each satellite - while now a satellite may be servicing Boise, Idaho, an orbit later it will fly over Los Angeles and will be swamped with traffic.
Expanding capacity is much more costly and risky. You have to plan a year or two in advance, book launches, pay hefty sums to sat manufacturers, launch providers, insurance companies, sweat about some doofus dropping your sat from a table. That doesn't mean land links are cheap - rights-of-way, and actual construction cost money, but much less than satellites. One must say a large scale satellite constellation has its advantages, especially if you can lower launch costs and create reliable satellite-to-satellite laser interconnections. If a huge company like Google goes into a price war with ISPs to win over part of the mobile user base, it may well establish a beachhead, drive some ISPs to bankruptcy and lock in many users.
Comms will be patchy - heavy rain and magnetic storms will bite into your link budget and sometimes block communications over whole continents. While sat-to-sat optical links will be unaffected, customers will suffer.
Alaska, Canada, Norway, Sweden, Greenland, Iceland, Finland, Russia, and the Antarctic penguins may be worse off after the switch - first because high-latitude connections need high-inclination satellites in a separate constellation (which is not profitable), and second because of aurora borealis/magnetic storms. (Please bear in mind this point is about small and mid-sized townships, not nomadic settlements/geological parties).
Security will be actually much worse - fiber optic cables are intercepted by advanced teams and state-level agencies, while satcomms are much more easily intercepted, jammed, and triangulated. Much like the mobile vs. fixed phones.
This said, here are some afterthoughts/nice tricks (that haven't been covered elsewhere) to improve link budgets and make a LEO satellite comms constellation more competitive:
There are usually many smartphones in a given urban locality. One can exploit that to do some heavy preprocessing (stealth bomber-style) to correct for multipath propagation and raise downlink speeds. The number crunching will have to be done in the satellite or, less likely, in the ground control center.
Again, many nearby emitters (smart phones) may kind of be made to coordinate with each other to act as a distributed phased array to get higher uplink speeds. This would make use of the fact that statistically not everyone is uploading a Youtube video (in normal situations).
I'm specifically not discussing latency - a variable well covered in other answers and comments. Nor do I touch upon the mass/size/energy budget of satellites in a global constellation.
Cubesats won't work. Because they're so small, you need powerful transmitters and receivers on the ground, plus large and complicated tracking dish antennas. If you want to use a small antenna on the ground, you need a large satellite.
Satellites have far less bandwidth (=how much data can it send per second) than a fiber cable. And you can upgrade a fiber cable to have more bandwidth easily (just replace the equipment at both ends, no new cable necessary).
A significant drawback to satellite based internet connectivity has nothing to do with the satellites but connection hardware, and the restrictions you'd have using the technology. Satellite communications requires line of sight, so you'd only be able to use a service like this if you had access to unobstructed sky. High-speed satellite communications requires a dish, and significant power requirements.
Any mobile devices using a satellite based internet service would be very limited in access and bandwidth - you'd have to be in clear space outside to use it and as you'd be stuck with directionless antennas your access speed would be very, very slow.
Wikipedia defines LEO as 100-1200 miles above the earth's surface. Let's assume 1,000 miles, since it's a nice round number, and we want to stay well clear of the atmosphere for fuel minimization.
Your minimum round trip distance is now 2,000 miles. If you previously had a fiber connection to a server 100 miles away, your latency has just degraded by an order of magnitude! And this assumes assumes a satellite directly overhead.
It is common for major websites to be hosted at multiple physical servers to cut down on latency - when I check BBC News from the East Coast of the US, I am not waiting for my packets to travel to Britain and back. The BBC has set up a mirror in the States - probably within a few hundred miles of my location.
Additionally, industrial grade routing equipment is big. A Google search for "industrial router" turned up this site: http://www.juniper.net/us/en/products-services/routing/ptx-series/ptx5000/
It lists the router as over five feet tall and weighing over 1,200 lbs - and it is plugged into grid power, and kept in a climate controlled room. If you start adding solar panels, altitude management, thermal control, RF comms equipment, radiation hardening - this is a multi-ton comms sat, not a cube sat. The market for routers this size is large enough to support several companies, each with several different models offered. There are likely many hundreds to thousands of routers this size currently in use.
You could leave the router on the ground, and turn the satellite into a "dumb" repeater - passing the packets on to other satellites and finally a ground station in a pre-set path, determined by the router on the ground. This exacerbates the latency issues. Now, any time a bit error or lost packet occurs, the satellite must send a message back through the chain all the way to the ground site to ask for a re-transmission - round trip minimum is now 4,000 miles.
I don't have stats to address power requirements, but I wouldn't be surprised if RF transmissions were an order of magnitude (or more) more energy expensive than fiber.
In the end, I think you've massively underestimated the amount of traffic that the Internet carries, the size of the hardware required to handle it, and the speed at which packets are required to move.