Tony Stark is a flirt (see Christine Everhart and Pepper herself in Iron Man, Natalie Rushman in Iron Man 2, and Maya Hansen in Iron Man 3, among others), like his father before him (see Peggy Carter in Captain America, and various women in Agent Carter).
You’re spot on about him wanting to be something of a father figure to Peter, but judging from the movie, it’s apparently at the level of wanting to maybe talk to the kid every few months and delegate most of the actual guidance to Happy Hogan, rather than wanting to start a romantic relationship with Peter’s legal guardian and become his actual step-dad.
As such, we never see any contact between Tony and May after that Civil War scene. In Spider-Man: Homecoming, he makes one reference to Aunt May’s attractiveness (in the ‘alibi video’ he makes with Peter), but immediately corrects himself as behaving inappropriately. Given his general level of flirtiness, this doesn’t seem contradictory to the later revelation that he and Pepper have worked things out post-Civil War.
The thread isn’t May/Tony; it’s “Aunt May is attractive”, and may have been a reference to the fan reaction at Marisa Tomei’s casting.