He never declared a political party, nor did he ever publicly endorse a political candidate, but he did privately endorse JFK. However, he tended to vote "the man" rather than "the party" in most national and local elections, some Democrats, some Republicans. He liked the Kennedys and LBJ, but also liked Eisenhower and Nixon. It is clear, however, that the very civil rights he was fighting for put him in direct opposition to the very, very racist Democrats in Washington, since it was those Democrats (with the only real exception being LBJ) who fought him the hardest every inch of the way.
He was not a Republican, but he leaned conservative on more issues than not, so to label him as a Republican isn't exactly accurate. But to label him a Democrat is far less accurate by any standards you choose to apply. Mainly, he tried very hard to not be partisan for partisan's sake. In 1956 he was offered a lot of money from both political parties to rally the black vote for each, a sum of money that, in his words, would have given him his first real-world possessions. But he turned both parties down, deciding not to "sacrifice [his] soul in the structure of partisan politics."
"I'm not here to tell you how to vote. That isn’t my concern. I’m not a politician. I have no political ambitions. I don’t think the Republican party is a party full of the almighty God nor is the Democratic party. They both have weaknesses. And I’m not inextricably bound to either party. I’m not concerned about telling you what party to vote for. But what I’m saying is this, that we must gain the ballot and use it wisely."
But for liberals... to have a friggin' Christian preacher be the leader of the Civil Rights Movement?!?! Uhm, no. Unacceptable.