Carson is a devout Christian and says a president's faith should matter to voters if it runs counter to America's values.
Responding to a question on NBC's "Meet the Press," Carson describes the Islamic faith as inconsistent with the US Constitution.
He says he "absolutely would not agree" that "we put a Muslim in charge of this nation."
Carson's comments come amid fallout over Republican front-runner Donald Trump's refusal last week to take issue with a man during a campaign event who wrongly called President Barack Obama a Muslim and said Muslims are "a problem in this country." Obama is Christian.