The 2019 Williams Institute Moot Court competition in Sexual Orientation and Gender Identity Law (in which I was a volunteer judge) revolved around this very question as well as whether the "ministerial exception" prevented an employment discrimination lawsuit by a bisexual employee under Title VII.
There is a current split between circuit courts on the question of whether gay people are covered under the 1964 Civil Rights Act, and thus the U.S. Supreme Court agreed to decide this question by consolidating Altitude Express v. Zarda and Bostock v. Clayton County, Georgia. In Zarda, the Second Circuit ruled that Altitude Express impermissibly discriminated on the basis of sexual orientation against Don Zarda, a sky-diving instructor (who was killed tragically in 2014 but the lawsuit has continued in his name by his surviving partner and family). Altitude Express's appeal of that ruling is being consolidated with an 11th Circuit ruling in Bostock which refused to do an en banc reconsideration of its holding that child welfare services coordinator working for Clayton County did not have a right to sue for sexual orientation discrimination and gender stereotyping.
Additionally, the Supreme Court also agreed to hear, R.G. & G.R. Harris Funeral Homes Inc. v. Equal Employment Opportunity Commission, a case of a transgender employee fired by a funeral home; the business is challenging the Equal Employment Opportunity Commission's decision that gender identity is covered under Title VII's sex discrimination ban. Today's Queer Quote is the issue in that case:
Whether Title VII prohibits discrimination against transgender people based on (1) their status as transgender or (2) sex stereotyping under Price Waterhouse v. Hopkins.
Price Waterhouse is a longtime precedent case (from 1989) which prohibited discrimination on the basis of sex (gender) stereotyping, ruling that this is covered under Title VII.
Presumably these will be some of the most significant cases of the 2019-2020 Supreme Court term, and be released as decisions in June 2020, right in the thick of the 2020 presidential campaign is heating up (hopefully both nominees will be known by then).
The doubt over whether LGBT people are protected by federal law against invidious discrimination based in sexual orientation and gender identity is why the Equality Act was introduced into Congress earlier this year.
Hat/tip to SCOTUSblog