I think Dr. Jones would be rather confused and guilty about his new father status...if Shia really does turn out to be his son, Indy would feel really bad about not being there for his kid when younger. Also, he would try to make up for his absense (in an attempt to not follow the ignorant example of his own father) by being as nice and accepting towards his kid as possible, which would make Indy Jr. feel overly important and would be annoyed. Indy would definitely have trouble bonding with his son, under the surprise circumstances of the secret coming out in this movie. Yeah, I think Indy would try too hard to be nice to his kid in an attempt to make up for not being there for Shia, and by film's end Jr. would eventually confess his true parentage...the son of Marion and Mack!!!
No, honestly, I think that would be a great twist to end the movie. Maybe Marion had conceived kid Indy all those years ago, and in order to protect Mack from Indy (who would surely be jealous of Marion being together with his partner), she lies and says that Shia is Indy's son. This would stay secret until the very end, and this revelation would make for a cool, funny twist ending! Think about it! It would also fix the problem of fans being insulted by a possible Indy-had-a-kid scenario.