A defiant Ben Carson returned to campaign in his home state on Wednesday, as the retired neurosurgeon continued to push back against the firestorm he ignited this week after saying that he did not believe a Muslim should be president.

Carson, the Republican presidential candidate gaining ground on the frontrunner Donald Trump in recent weeks, told reporters that "political correctness" bears the responsibility for the criticism he has faced since Sunday, when he said he would "not advocate that we put a Muslim in charge of this nation".

After initially standing by the comments, the Detroit native has begun an attempt to partly walk back his remarks, something he continued on Wednesday at a press conference in Jackson, Michigan, in front of a plaque commemorating the spot of the GOP's birthplace in 1854.

"We have an American culture and we have an American constitution and anybody who's going to occupy our White House should be living in a pattern that is consistent with our constitution and our culture," Carson said.

Surrounded by supporters wearing "I'm with Ben" stickers, the Tea Party darling also said he would push a six-month amnesty for US corporations holding money in overseas account where taxes are lower. If the funds are turned to US soil, "it wouldn't cost you anything", he said.