Speaking on Sunday on CBS’ programme Face the Nation, the vice president quoted from Dr King’s “I Have a Dream” speech as he defended Mr Trump’s latest pitch to secure funding for a barrier along the United States’ southern border.
The budget impasse over the wall has resulted in the longest government shutdown in history, one that has left thousands of federal workers without pay.
“One of my favourite quotes from Dr King was, ‘Now is the time to make real the promises of democracy’,” Mr Pence said.
“You think of how he changed America. He inspired us to change through the legislative process, to become a more perfect union.”
“That’s exactly what President Trump is calling on Congress to do,” Mr Pence added.
“Come to the table in the spirit of good faith. We’ll secure our border. We’ll reopen the government, and we’ll move our nation forward as the president said yesterday to even a broader discussion about immigration reform in the months ahead.”
Mr Pence was referring to the president’s latest proposal to end the partial government shutdown. On Saturday, Mr Trump offered Democrats three years of deportation protections for some immigrants, including many who were brought to the country illegally as children, in exchange for $5.7bn in wall funding.
Democrats, who have long sought protections for beneficiaries of the Deferred Action for Childhood Arrivals programme, who are known as “Dreamers,” immediately rejected the proposal, arguing that the president should reopen the government before any talks on border security begin. Conservatives saw the president’s proposal as amnesty.
On Sunday, Mr Pence appeared on shows and defended Mr Trump’s proposal, which he said “provides a framework” for ending the shutdown.
“What the president presented yesterday really is an effort to bring together ideas from both political parties,” Mr Pence said on CBS.
“I think it is an act of statesmanship on the president’s part to say, ‘Here’s what I’m for. It includes my priorities’.”
Mr Pence’s comments about Dr King drew immediate pushback from Democrats, including Representative Jackie Speier, who said Mr Trump “is no MLK”.
“To equate the legacy of one of America’s finest statesmen and champions of civil rights with a vanity project built on racist ideology and hatred is beyond disgraceful,” Ms Speier tweeted.
The National Association for the Advancement of Coloured people (NAACP) called Mr Pence’s comments “an insult” to King’s legacy.
The partial shutdown has closed parks across the country, including the Martin Luther King Jr National Historical Park in Atlanta. The park reopened in time for the holiday after it received a grant from Delta Air Lines.
One of Dr King’s children, Martin Luther King III, talked about the reopening on CNN on Sunday.
The Washington Post was unable to reach Martin Luther King III for comments about Mr Pence’s remarks.
In recent years, the civil rights leader’s children have asked Mr Trump to stop using what they believe to be divisive rhetoric.
Martin Luther King III called out Trump last year over reports that the president used vulgar words to deride African countries and said the country should instead bring more immigrants from countries such as Norway.
“When a president insists that our nation needs more citizens from white states like Norway, I don’t even think we need to spend any time even talking about what it says and what it is,” Martin Luther King III said, according to the Associated Press. “We got to find a way to work on this man’s heart.”
During the holiday commemorating Martin Luther King Jr last year, his daughter, Bernice King, also told the AP that she hopes Mr Trump would observe the holiday by avoiding offensive tweets.
“If he can dare to do that, I would be proud on that day that our president honoured Dr King by not doing things that are offensive,” Bernice King said.