Not quite. Michelson-Morley showed ether was not necessary to explain the observations. They never said it didn't exist.
I disagree. The Michelson-Morley experiment was done at opposite ends of the earth's orbit (6 months apart). The results were identical. If there were an ether that light traveled in, then the speed of light would have been different. Hence, the ether theory was discredited. (A good textbook on the subject is "Special Relativity," by Anthony French. MIT Press.