I think you may be making the mistake of applying modern controversies to a different era. During his lifetime the subject of gay marriage was not really an issue of public controversy.

The earliest polling that I can find on gay marriage is from 1988, twenty years after MLK was killed, that found only 11% of Americans supported gay marriage: https://www.gazettenet.com/Archives/2015/03/gaymarriage-hg-030615

There's no reason to believe that any significant proportion of people in the US in the 60s supported gay marriage, and historically African Americans and Southerners were less likely than the rest of the population to support gay marriage. While there is no way to know his opinion on a subject that wasn't much discussed at the time, it would have been very rare for a person in his shoes to be supportive of gay marriage.

There are disagreements amongst his surviving relatives over whether or not he would have supported gay marriage if he was alive at a time when it was a matter of public controversy.