One unforeseen consequence of Mr. Trump’s dominance of the Republican Party has been how thoroughly he shattered the expectations of religious conservatives — the cornerstone of the party’s base — for a president’s conduct and leadership. The mold for appealing to those voters used to look a lot like Mr. Pence: an evangelical Christian who has unabashedly taken up causes dear to the activist right, like opposition to abortion and gay rights and dismantling government regulations on businesses, schools and churches.

One of the biggest questions hanging over Mr. Pence’s future is whether Republican voters in a post-Trump world will embrace a candidate who seems to embody what the party thought it needed in a nominee before Mr. Trump came along, and who lacks the fearlessness and audacity that many conservatives say Mr. Trump demonstrated as he pushed policy and the courts further to the right than any president in recent history.

“There are some qualities that Mike Pence has that I’ll continue to hope and pray that Donald Trump adopts — and the reverse is true as well,” said Tony Perkins, the president of the Family Research Council, a pro-Trump, social conservative group. “There are qualities that Donald Trump has that I hope and pray Mike Pence adopts. And that is that he picks up a little bit of the fight and courage that Donald Trump shows on a daily basis.”