No, chaste can apply to married sex just as to unmarried abstinence. As I explained in an earlier comment:
chastity means different things depending on one’s state in life.
For the unmarried to be chaste is to be totally sexually abstinent.
For the married, to be chaste is to practice sex chastely, properly, in purity—which means selflessly rather than selfishly. One can use one’s spouse selfishly, which is unchaste. One can give oneself to one’s spouse selflessly (and both can do this mutually) and be chaste in one’s sexual relations.
Chastity applies to everyone; everyone is to do the right thing regarding sex. For the unmarried, the right thing is total abstinence and that’s being “chaste.” For the married, doing the right thing is to be selfless, not merely selfishly self-gratifying.
Some people use “chaste” when they mean “abstinent” or “continent,” just as some people use “celibate” when they mean “continent” or abstinent. A lot of Catholics misuse these words. It would really help to avoid confusion if people would use “celibate” and “chaste” correctly.
Chastity - abstaining from sexual relations.
Source: Webster's Unabridged Dictionary 2nd Edition 1979
This is why there is a vow of celibacy and a vow of chastity.