I'm not sure Indy was expressing Republican support when he uttered "I Like Ike". I think his retort to Spalko was just a smart-alec response to indicate his allegiance to the US and his opposition to Communism using a pop-culture line from the era. He certainly doesn't appear to be supportive of the administration's witchhunts or the way in which the government has treated him in the past. Plus it's a lot snappier than saying "I object to your actions because I'm an American Citizen and embrace democracy"
Indy is an academic and free-thinker. I don't think he'd have much regard for the government of the day...