What did those who supported manifest destiny want to do about Mexico after the war?

Answer:
They were divided as to what to do with Mexico. Some proponents argued in favor of annexing all of Mexico; however the vast majority were against, due to the fact that by annexing Mexico, the United States would have had to grant US citizenship to millions of Mexicans. This was badly seen, due to racial reasons. Senator John Calhoun of South Carolina made these views clear in a speech to Congress on January 4, 1848:

We have never dreamt of incorporating into our Union any but the Caucasian race---the free white race. To incorporate Mexico, would be the very first instance of the kind, of incorporating an Indian race; for more than half of the Mexicans are Indians, and the other is composed chiefly of mixed tribes. I protest against such a union as that! Ours, sir, is the Government of a white race.... We are anxious to force free government on all; and I see that it has been urged ... that it is the mission of this country to spread civil and religious liberty over all the world, and especially over this continent. It is a great mistake.
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