Well no, there is never even a point where they mention he's gay.
Not as such, but if Regis tries to find his off switch during the restaurant scene, he does seem awfully pleased and enthused about the sudden invasion of personal space. Perhaps he's just generally excitable...
ETA: The dialogue in question (after using the CI-Splitter on Stepford)
Stepford[startled]: Oh my, Dr Regis! How forward!!
Stepford[sly smile]: I had no idea you felt this way.
Regis[perplexed]: Uh...
Stepford[still smiling]:We must discuss this later in private.
Regis[awkwardly]: ...I'm... just going to..go..uh...
Stepford[conspiratorially]: Of course! Of course.