What does immigration have to do with funding for the Department of Homeland Security? According to some Republicans, a whole lot. Here’s a breakdown of what’s happening in the fight to keep the DHS budget running. (Pamela Kirkland/The Washington Post)

Congressional Republicans on Tuesday continued their assault on President Obama’s signature accomplishments and his ambitious budget proposal. But they faced stiff resistance from Democrats.

The developments illustrated how a GOP majority in both chambers is no guarantee that Republicans will get what they want.

The Republican-led Senate tried — and failed — to move ahead on a bill that would fund the Department of Homeland Security but strip it of money for Obama’s executive actions on immigration. The Republican­controlled House again voted to repeal Obama’s sweeping health-care reform law, a symbolic move that stands no chance of taking effect. And in both chambers, Republicans sought to shoot down Obama’s $4 trillion budget in advance of releasing their own plans.

Senate Democrats blocked a $40 billion DHS funding bill that passed the House and would cover the department through September. They oppose the bill because of the GOP’s effort to stop Obama’s executive actions on immigration, including his decision to stem the deportations of millions of undocumented immigrants.

The move leaves unresolved whether DHS will receive a new funding bill before current spending runs out Feb. 27. The agency is bracing itself.

The U.S. Capitol Dome silhouetted against the rising sun in Washington, D.C. in February 2010. (JEWEL SAMAD/AFP/GETTY IMAGES)

“A shutdown of the DHS in these times is frankly too bitter to contemplate, but we have to contemplate it,” said DHS Secretary Jeh Johnson.

The vote to proceed to debate on the bill fell almost entirely along party lines, with all 46 Democrats uniting to leave Republicans short of the 60 votes needed to advance the measure.

Sen. Dean Heller (Nev.), whose state is home to many Hispanic residents, was the sole Republican to join with Democrats. Majority Leader Mitch McConnell (R-Ky.) voted no, but only as a procedural tactic that allowed him to bring the bill up again.

Minority Leader Harry M. Reid (D-Nev.) told reporters that a clean DHS spending bill — meaning one without provisions regarding Obama’s immigration actions — would be the only measure Senate Democrats could support.

Ahead of the vote, Democrats claimed Republicans were risking shutting down a key government agency amid heightened concerns about terror threats. Republicans countered that Democrats were acting unreasonably by preventing even an initial step forward on the measure.

“It’s time for Congress to be resolute in its determination to defend America,” said Sen. Richard J. Durbin (Ill.), the second-ranking Democrat, after leaving a caucus lunch with Johnson.

Sen. Roy Blunt (R-Mo.) said: “This is just a vote to debate the bill. Democracy doesn’t work if you don’t debate.”

On the other side of the Capitol, the House took aim at Obama’s health-care law by passing legislation to repeal it. It was the 56th time it has voted to repeal all or part of the bill.

But the president will not sign any bill that seeks to undo the law known as “Obamacare.” Democrats criticized the GOP’s push to shred the law.

“The clock is ticking on the bill for Homeland Security. That’s our responsibility — to support and protect. Let’s get about the business that we take an oath to do, instead of, for the 56th time, bay at the moon,” said Minority Leader Nancy Pelosi (D-Calif.) in a House floor speech.

Republicans, meanwhile, declared Obama’s budget plan to be a legislative non-starter as they criticized the new taxes it would require to pay for a slate of new programs.

“It’s dead on arrival,” said Sen. John McCain (R-Ariz.).

Sen. Tim Scott (R-S.C.) called it “unrealistic — completely.”

Obama’s plan, which includes proposals for new early education programs and free community college tuition, would require new tax revenue from wealthy Americans and large corporations to fund the initiatives.

“There’s no greater contrast than showing what this new American Congress is for and what the president supports. His new budget will give the federal government an 11 percent raise by taking more out of the economy in taxes,” House Majority Leader Kevin McCarthy (R-Calif.) told reporters.

Republicans are expected to unveil a more austere budget proposal in the coming weeks. That will launch a protracted process that will put the GOP under intense pressure to craft a budget plan that can win the president’s signature by the start of the next fiscal year in October.

One bill that congressional Republicans are poised to move ahead with even without much support from their Democratic colleagues would approve construction of the Keystone XL oil pipeline. McCarthy said Tuesday that the House plans to vote on the Senate-passed bill next week.

But even that measure has its limits. Obama has vowed to veto it, and Democratic leaders have said they will uphold his veto.

Greg Jaffe contributed to this report.