Semi-submersibles--which is what Sea Launch was (former Ocean Odyssey)--are pretty stable. The marine risers (i.e. the big pipes that connect to the subsea blowout preventer) don't tolerate a lot of angle. Source: Used to work on, among other things, offshore drilling rig mooring analysis.

But I can imagine all sorts of reasons why using an offshore platform introduces a bunch of challenges that launching from land doesn't have. Though of course, the ability to get away from people is a plus too.