Yes he was exactly that kind of person.American sectarianism persisting to this day always intrigued me as an idea, though even if Graham was personally anti-Catholic I doubt he'd focus on it too much as President. He doesn't really seem to be the type of person to act like that.
His entire political involvement in 1960 was to stop the papist. He was also Anti-Semitic and a horrific racist. And never gave up his homophobia. He just had better PR then his peers. He would absolutely push it and the rest of his horrible views.