From THE MAKING OF STAR TREK (p.254):
"Christine Chapel is well-educated for her task, with a doctorate of her own in Bio-Research. (sic)"
The character sketches/biographic details in TMOST were taken from the second season Writer's Guide, and thus, reflective of the thinking in the TOS production office ca. 1967. That bit of information never made it into dialogue onscreen, so technically, isn't "canon," but the concept of Chapel having a doctorate of her own - albeit not an MD, probably a PhD or DSc - wasn't pulled out of thin air.
Later, she got her MD prior to TMP.
[Edited to fix an autocorrect-induced word substitution error.]