No, it's not. It is a scientific fact that one cannot change their chromosomes, but the scientific term sex long predates our understanding of genetics. You can redefine sex to be wholly chromosome-based if you want, but it's reductionist.
Stem cell and bone marrow transplants actually can change, or at least add to, the DNA and chromosomes present in someone, too.
Basically: gender and sex in biology and medicine are a lot messier than the neat pre-existing linguistic and social categories we've tried to back-fit them onto.