The governor has no more authority than the supreme court to issue directives that are contrary to existing laws. If the governor wants to change a law he must persuade the general assembly to send him a bill that produces the desired result.
It’s not just statutes, but the Kentucky Constitution as well
Section 233A
Only a marriage between one man and one woman shall be valid or recognized as a marriage in Kentucky. A legal status identical or substantially similar to that of marriage for unmarried individuals shall not be valid or recognized.
http://www.lrc.ky.gov/legresou/constitu/233A.htm