The Texas law went into effect on Wednesday when the 5th US Circuit Court of Appeals granted the state's request for a stay of a district judge's injunction blocking the law.
The law forbids social media companies with more than 50 million active users per month from banning members based on their political views and requires them to publicly disclose how they moderate content.
It was signed into law by Texas Governor Greg Abbott, a Republican, in September.
Internet lobbying groups NetChoice and the Computer & Communications Industry Association filed a lawsuit against the measure, and US District Judge Robert Pitman in Austin, Texas, issued a preliminary injunction in December.
Pitman had found that the law would harm social media companies' free speech rights under the First Amendment of the US Constitution.
The tech groups, in their emergency request, asked the Supreme Court to "allow the District Court's careful reasoning to remain in effect while an orderly appellate process plays out." © Reuters