While Gorsuch has never ruled directly on LGBT rights, more than enough of his judicial history and philosophy is known to reveal he's no friend of the LGBT community. As a result, for the first time in history, the LGBT-rights organization Lambda Legal is opposing his nomination (link here).
Make no mistake about it, Gorsuch was actually selected by the ultra-conservative, LGBT-hating Heritage Foundation. Trump long ago freely admitted on camera and more than once that he was letting that organization pick all final candidates and then he'd make the choice only from their list.
In turn, the Heritage Foundation conducted extensive closed-door interviews of all known conservative friends and associates of various candidates to learn more about their private opinions and comments on civil rights and other issues. As a result, no LGBT-friendly candidates could end up on their final list.
Democrats in the Senate have vowed to block any unacceptable Trump nominee, given Republicans blocked President Obama's nominee for a full year and never even convened hearings on the choice. Hence, Democrats are only giving Republicans a taste of their own medicine.
Supreme Court nominees can still be filibustered with only forty votes, and it's essential Democrats and Independents hold the line and refuse to allow Gorsuch's confirmation. If you live in a state with a Democratic or Independent senator, contact them every day and urge them to block Gorsuch (contact details here).
Gorsuch could be the vote that reverses same-sex marriage rights; his ultra-conservative record suggests that may be possible, as well as other draconian votes against fundamental civil rights. He must not be confirmed.