The measure allows one spouse to adopt the other's adopted or biological children.
The majority leader told reporters that the Employment Non-Discrimination Act has "a chance" of getting a vote in the Senate this year.
At the end of the legislative session, Republican lawmakers in Texas are trying to quietly pass bills that activists say are transphobic and antigay.
The Nevada assembly passed legislation Tuesday to address hate crimes against transgender people.
The U.N.'s Office of Human Rights has a message for LGBT people around the world.