WASHINGTON — Republican victories in crucial Senate and governors’ races this week have tightened social conservatives’ grip across American government, strengthening the party’s power as it seeks to limit abortion rights and push harder to the right on a number of divisive cultural issues.

Even as Democrats captured the House and promised to act as a check on President Trump and Republican policy priorities, conservatives were breathing a deep sigh of relief on Wednesday after strengthening their majority in the Senate. Their gains in the upper chamber could have a far-reaching impact on the remainder of the president’s term, particularly on his ability to continue shifting the ideological balance on the federal courts.

The election of conservatives to Democratic-held Senate seats in Indiana, Missouri and North Dakota — as well as to an open seat in Tennessee, and possibly in Arizona, Florida and Mississippi, where races have not been decided — will almost certainly add to the Republicans’ one-seat majority. It will also dilute the votes of moderate Republicans like Senators Susan Collins of Maine and Lisa Murkowski of Alaska, both of whom are abortion rights supporters who have opposed judges on the grounds that they could threaten Roe v. Wade.

President Trump, who has already filled vacancies on the federal bench at a faster pace than most previous presidents, and cemented a conservative majority on the Supreme Court with two appointments so far, is now in a stronger position to put forward even more conservative judicial nominees if he chooses to.