Anyone who commits any aveira, Torah or rabbinic is disqualified (Choshen Mishpat 34,1-3) until he is punished by a bet din or does teshuva (ibid seif 29). This, in fact, was the basis for Rav Moshe disqualifying non-Orthodox weddings. However, a person must be disqualified by a bet din after testimony regarding him.which means two, not ten (ibid seif 25). It would seem though that according to Rav Moshe there would be an anan sahadei, which works like official testimony, where it is known that people are connected to certain non-Orthodox groups.
The distinction between someone who is mechalel Shabbat in public and in private only refers to treating him like a gentile (e.g. rendering his wine non-kosher).