Arnold was more embarrassed of that day for his ego and family then he was when the fat Burrito came out and said she had a kid with arnold
I really don't think Arnold would have a problem with one of his sons being gay. There have always been lots of gays in his life (and I'm not talking about Weider, how about Joe Gold and his partner Chuck Pendleton, who Arnold was good friends with, so much that he was the one who spread their ashes together off Marina Del Rey)
(but yes, it's one of those open secrets in hollywood that Taylor Lautner is very very very gay)
Here is a pic of Arnold in the late 70s, off the juice, trying to have a more normal physique with the hope of becoming a legitimate actor.