WASHINGTON—The Democratic-controlled House approved legislation Thursday that would make Washington, D.C., the 51st state, but the bill faces an uphill battle in the Senate amid broad Republican opposition and hesitation from some centrist Democrats.

Proponents cast the bill as a civil-rights priority, saying the city’s residents deserve full self-rule and representation in Congress. Critics call it a power grab by Democrats to gain two more senators and point to possible constitutional hurdles.