The linked blogpost in the question is from a series of Vintage Space posts in Discover Magazine. Amy Shira Teitel also has an extensive series of informative and enjoyable videos on her Vintage Space YouTube channel.
The video Sea Dragon is the Massive Rocket of Spaceflight Dreams is worth watching for a helpful breakdown of the Sea Dragon proposal. I've transcribed her discussion of the recovery of both the first and second stages between
07:19. The last paragraph indicates that Aerojet asserts that the proposed Sea Dragon implementation would be a series of solvable engineering challenges.
The first and second stages were both recovered at first just using just simple aerodynamic braking as they fell through the upper atmosphere, but the active stage of recovery would be using an aerodynamic deceleration device.
This decelerator was a reliable, and very easy system. It was a large conical flare 300 feet in diameter, that could be pressurized with the same pressurizing gas as was used in the pressure-fed engines for each stage. This also ensured that both stages would hit the water at the correct nose-down orientation, a way that would minimize the forces and stresses on the body such that it could be refurbished easily and reused, as opposed to building a new one. After the first and second stages both splashed down, they would be recovered and towed back to the assembly lagoon. There they would be refurbished, and any parts that need replacing could be replaced. But at the end of the day, most of the rocket could be just refurbished, mated together again in that lagoon, and used again on another Sea Dragon launch.
As Aerojet noted in its 1963 report, the sea dragon definitely represented a massive leap in technology, but it wasn’t one that was completely impossible. Every challenge with Sea dragon was actually just an engineering challenge, and engineering challenges could be solved.
The report NASA-CR-52817 Sea Dragon Concept. Volume 1: Summary (Aerojet-General Corp.) N88-71080m 28-Jan-1963 discusses recovery of both stages in section III-C, and the second stage in particular on pages III-c-19 to 21.
Additional discussion of recovery development can be found throughout this report dated a few weeks later.
The second-stage auxiliary engines provide orbital injection thrust for the payload, and the second stage will go into
orbit with the payload. During ascent into orbit and after the second stage main engine has ceased firing, the expandable nozzle skirt will be separated and ejected from the stage. After attaining orbital condition, the payload and expended second stage will be separated. When the second stage has reached the desired position, small retro rockets in the nose will be fired to eject the vehicle out of orbit. A velocity impulse of 480 ft/sec will be required to give an initial re-entry angle of 2.5 ° from the local horizontal.
Aerodynamic stabilization and deceleration of the stage will be attained with an inflatable flare similar to the one used for first stage recovery. The size of the flare will be smaller than the flare used on first stage. The available tank pressure is lower, 50 psi, however, the recovered weight is also lower, 1.2 x 10E6 lb. This 50 psi tank pressure, available for axial load reactions,corresponds
to a terminal velocity of 210 ft/sec for which a flare diameter of 240 ft is required.
Aerodynamic heating for second stage recovery will be more severe than for first stage. Use of flare material with higher thermal resistance_ such as Rene 41 mesh and ablative coatings will be necessary.