WASHINGTON — The consequences of President Trump’s nomination of Neil M. Gorsuch to the Supreme Court — and the Republican blockade of President Barack Obama’s nomination of Merrick B. Garland in 2016 for that seat — became powerfully clear on Tuesday after the court’s conservative majority handed down major decisions to uphold Mr. Trump’s travel ban and in favor of abortion rights opponents.

Social conservatives cheered the court’s ruling that a California law requiring “crisis pregnancy centers” to provide abortion information likely violates the First Amendment. Some conservatives also viewed the ruling — their latest win to advance their anti-abortion cause since Mr. Trump has taken office — as another opportunity to energize their base ahead of the November elections.

The travel ban decision drew more conflicting reactions from conservative voters and religious groups, with some criticizing it as anti-immigrant. Several groups supporting immigrants deemed the travel ban decision “shameful” and “hateful.” And many Democratic leaders denounced both rulings.

What many partisans on both sides agreed on, though, was that Justice Gorsuch — who voted with the 5-to-4 majorities in both cases — was an especially key figure in Tuesday’s decisions, because he wouldn’t have been on the court if Mr. Obama had been successful with the original nomination of Judge Garland.