I don't know who would be in charge of doing the requiring (perhaps FCC [edit: err... FAA] with American companies / launches from America?), but I can't imagine that anybody would be willing to not use pressure suits during the risky portions of missions. Soyuz 11 was really bad.. nobody is keen on repeating it.
Thankfully I think that safety is something that economic pressure will address really well with commercial spaceflight. High-profile accidents didn't sink either the American or Russian national space programs but I think that at least in the foreseeable future commercial outfits will feel a lot more vulnerable . A high profile deadly accident for SpaceX would be devastating for them; at least moreso than an accident for a government agency with some strategic military purpose partially behind it. I expect commercial spaceflight companies to take safety at least as seriously as government agencies (at least while they are still in their infancy. Once SpaceX becomes the real-life "Union Aerospace Corporation" that might change. ;))