WASHINGTON—The Supreme Court on Thursday ruled that Texas could reject a request for a specialty license plate featuring the Confederate battle flag, saying state-issued auto tags aren’t an open forum for any message a sponsor might want to promote.

The 5-4 vote settled a free-speech battle that has divided the federal appeals courts over what modern license plates principally are: an official document displaying government-approved information, or a public forum where any party that can pay the fee is free to communicate...