Updated on Aug. 10 at 4 p.m. ET

President Trump is continuing to voice his frustration with Senate Majority Leader Mitch McConnell, tweeting on Thursday that the Kentucky Republican should "get back to work" after last month's failure to pass a health care alternative to the Affordable Care Act.

The president's latest tweets calling out the Kentucky Republican came as Trump himself is taking a "working vacation" at his own golf resort in New Jersey.

And pressed by reporters after a meeting with Vice President Pence later on Thursday as to whether McConnell should step aside as GOP leader, as some conservative commentators have argued, Trump wouldn't give a clear yes or no.

"If he doesn't get repeal and replace done, and he doesn't get taxes done — meaning cuts and reform — and if he doesn't get a very easy one to get done, infrastructure, if he doesn't get them done, then you can ask me that question," Trump said.

"I just want [McConnell] to get repeal and replace done. ... And all I hear is 'repeal and replace,' and then I get there and I said, 'Where is the bill? I want to sign it, first day.' And they don't have it," Trump said, signaling he was ready to move on to other key agenda items.

"They should have had this last one done. They lost by one vote," the president continued, talking about the Senate's failure last month to move forward on efforts to repeal Obamacare. The White House has confirmed that Trump and McConnell did speak this week and talked about health care.

But on Thursday, Trump doubled down on the Senate's narrow failure, saying that, "For a thing like that to happen is a disgrace. And frankly it shouldn't have happened. That I can tell you. It shouldn't have happened."

Trump's heightened criticism on Thursday toward McConnell — with whom he has had a sometimes strained relationship — came after McConnell first threw down the gauntlet earlier in the week over the GOP's health care collapse. Speaking back home in Kentucky, the Senate veteran argued it was Trump's political inexperience that had led him to set "excessive expectations" over wanting to speedily repeal Obamacare.

"Part of the reason I think that the storyline is that we haven't done much is because, in part, the president and others have set these early timelines about things need[ing] to be done by a certain point," McConnell said on Monday, according to ABC News.

"Our new president, of course, has not been in this line of work before. And I think he had excessive expectations about how quickly things happen in the democratic process," the Senate majority leader continued. "So part of the reason I think people feel we're underperforming is because too many artificial deadlines — unrelated to the reality of the complexity of legislating — may not have been fully understood."

Unsurprisingly, that explanation didn't sit well with the commander in chief. On Wednesday, Trump fired his first missive at McConnell, asking, "After 7 years of hearing Repeal & Replace, why not done?"

Dan Scavino, Trump's social media director, had shared similar sentiments earlier from his personal account.

And Fox News Channel host Sean Hannity, a fervent Trump backer, called McConnell "weak" and "spineless."

Not only is it highly unusual for a sitting president and his allies to attack a Senate leader of his own party, but Trump himself did very little to rally any support for the GOP's health care alternative, which was rushed out and voted on in the middle of the night — a process that drew condemnation from members who even voted for it.

McConnell's office hasn't responded to Trump's latest criticisms this week, but Sen. Orrin Hatch, R-Utah, offered up his support for the majority leader.

Still, the escalating tension doesn't bode well for the rest of the Republican legislative agenda in the Senate when Congress returns to Washington, D.C., next month. Lawmakers face a daunting list that includes a tax overhaul, raising the debt ceiling and more.

Another thing making the Trump-McConnell spat complicated and even more surprising — McConnell's wife, Elaine Chao, is the president's secretary of transportation.

But it wasn't just the White House that was, at least initially, pressing for an ambitious legislative agenda.

"The two biggest issues we're moving forward with in the first half of the year obviously are repeal and replacing Obamacare and tax reform," McConnell said back in January at the GOP congressional retreat.

The latest jab from the president toward the majority leader comes just after Trump gave McConnell a political win on Tuesday night, when the president threw his support behind Sen. Luther Strange, R-Ala., ahead of a primary next week in a special election race.

Strange, who was appointed to fill now-Attorney General Jeff Sessions' seat, has the backing of McConnell, the majority leader's superPAC and the National Republican Senatorial Committee.

The endorsement was a blow to Rep. Mo Brooks, who is fighting to earn a spot in a runoff as a result of Tuesday's voting — with both Brooks and Strange likely to finish behind Roy Moore, a controversial former Alabama Supreme Court justice best known for refusing to remove a Ten Commandments memorial from a state judicial building. Brooks was critical of Trump last year during the presidential campaign, pointing to his "serial adultery," and more recently, the congressman slammed the president for his "public waterboarding" of Sessions on Twitter.

On Wednesday, though, as Trump's feud with McConnell was escalating, Brooks tried to seize an opening, urging the president to reconsider his endorsement since Brooks wants to oust McConnell from the GOP leadership.

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