WASHINGTON — Congressional Republicans greeted President Trump’s first full budget on Tuesday with open hesitation or outright hostility. But it was not clear that they could come up with an alternative that could win over conservatives and moderates while clearing a path for the tax cuts and policies they have promised for years.

The budget battle ahead mirrors the continuing health care fight, in which concessions to Republican moderates alienate conservatives, while overtures to conservatives lose moderate votes. But with Republicans in full charge of the government, the onus is on their leaders to reach a budget agreement in a matter of weeks that would ease passage of the president’s promised tax cuts as well as a new spending plan that would reshape the government in a Republican mold.

“It is now up to the Congress to act,” Mr. Trump said in his Budget Message. “I pledge my full cooperation in ending the economic malaise that has, for too long, crippled the dreams of our people. The time for small thinking is over.”

Mr. Trump’s $4.1 trillion budget, with its deep cuts to poverty programs, biomedical research, student loans and foreign aid, will not pass, as Republicans on Capitol Hill have freely acknowledged and even the White House is aware. Republicans on Capitol Hill parted ways with the president not only on many of his deepest cuts but also on some of his smaller proposals, like resurrecting a national nuclear waste repository in Nevada and ending the Great Lakes cleanup program.