"You notice that he didn't specifically say what exactly he was objecting to," the president said when asked about the suit by ABC News Chief Anchor George Stephanopoulos during the interview, which aired Friday on Good Morning America.
"I'm not going to apologize for trying to do something while they're doing nothing," the president added later.
Boehner said Wednesday during the announcement of the lawsuit that the president had "not faithfully executed the laws," but Obama sharply disagreed during his interview with ABC News.
"The suit is a stunt," he said.
As for immigration overhaul, the president cited public support for it, insisting he was open to compromise with the Ohio Republican during his interview with Stephanopoulos.
"What I've told Speaker Boehner directly is, 'If you're really concerned about me taking too many executive actions, why don't you try getting something done through Congress?'" the president said.
"You're going to squawk if I try to fix some parts of it administratively that are within my authority while you're not doing anything?" Obama said, directing his comments toward Republicans.
Obama Says Country Under Threat His Entire Presidency
During the wide-ranging interview, Stephanopoulos asked the president about how serious of a threat ISIS -- the radical Sunni Islamic group fighting to take control of Iraq -- posed to the United States, citing a recent op-ed by former U.S. Ambassador to Iraq Ryan Crocker, who served under both Obama and Republican President George W. Bush.
In the Washington Post, Crocker warned the U.S. "would be foolish to think that ISIS will not plan attacks against the West now that it has the space and security to do so."
Said Obama: "I think we have been under serious threat my entire presidency. And we have been under serious threat predating 9/11 from those who embrace this ideology."The president did concede that the group has grown more powerful in "some places" when asked about the issue by Stephanopoulos.
"Well, they're gaining strength in some places. But we've also got a lot better at protecting ourselves," the president added.
The president announced earlier this month that he would send up to 300 military advisers to Iraq to assist the country's government-backed forces as they continue to clash with ISIS, which has already taken control of several cities and towns within the country currently headed by Shiite Prime Minister Nouri al-Maliki.
Obama: Public Doesn't Have to Support Foreign Policy 'Every Minute'
During the interview with ABC News, the president was asked about a specific passage in the book he wrote in 2006, "The Audacity of Hope," in which he, referring to Bush's foreign policy, cites the importance of having a strategy that the public supports.
Now, with polls showing the majority of Americans are not in support of the president's foreign policy, Stephanopoulos asked whether he is failing by his own standard. Obama cited support he had during his first term in the White House and said the public didn't necessarily have to support him all the time.
"Not at every minute George, not every week because there are going to be times where the world is messy," he said
"What I've learned is I can't operate on a daily news cycle or a weekly news cycle. One of the things you also realize during the course of five years is, if the problems were easy, somebody else would have solved them," he said.
"And one of the great challenges of this job, but one of the great privileges of this job is that you know you're tackling stuff that is really tough," the president added. "And I'm glad that after five years, I'm still here and able to do it."