(CNN) All along, the White House had hoped that President Donald Trump's torment over Russia would end with Robert Mueller 's final report.

Now the special counsel's final flourish is beginning to look like just the start of the President's frustrations.

House Democrats, armed with subpoena power, on Wednesday announced a broader-than-expected investigation into Trump's ties to Russia, Saudi Arabia and just about anywhere else. And criminal probes are digging ever deeper into the global business record of the real estate billionaire who became President and therefore opened himself to painful scrutiny.

Trump's red line -- that no one should delve into his or his family's personal finances beyond a relationship with Russia -- has been snapped.

So it's no wonder that he offered a glimpse of seething inner fury at the State of the Union address on Tuesday night, coining a Nixonian couplet to deliver Democrats a warning that worked on multiple levels.

"If there is going to be peace and legislation, there cannot be war and investigation," the President said. "It just doesn't work that way."

A testy back-and-forth between House Democrats and the President on Wednesday offered a taste of an ugly confrontation that is all but certain to ensue with the finalizing of the Mueller investigation, possibly within months, and as multiple House committee inquiries targeting the President and his circle lumber into action.

House Intelligence Chairman Adam Schiff announced Wednesday that he would expand his scope beyond Russia to examine whether Trump's business interests are influencing his foreign policy decisions.

The California Democrat said he would look at any "credible allegations of leverage by the Russians or the Saudis or anyone else."

The revelation came a day after CNN first reported that prosecutors in New York wanted to interview Trump Organization executives after opening at least two investigations into Trump-related entities, including possible campaign finance violations relating to hush money paid to women by Trump's former personal lawyer Michael Cohen.

The new interest in Trump's past and present business holdings confirms one of the worst fears of the President's confidants: that even if Mueller gives him a pass, he faces prolonged and perilous exposure to new investigations that could last for months. A President who has not enjoyed a single day free of the shadow of investigations could potentially be under a legal and political cloud -- on issues wider than Russia -- for the rest of his term.

JUST WATCHED See Trump's sarcastic response when asked about Schiff Replay More Videos ... MUST WATCH See Trump's sarcastic response when asked about Schiff 00:46

Trump did not contain his frustration when asked about Schiff's move.

"He's just a political hack trying to build a name for himself," the President told reporters, adding that Schiff had no "reason to do that." He kept up his complaints on Twitter early Thursday morning, bemoaning "unlimited presidential harassment."

Schiff, a former prosecutor, responded on Wednesday with a cutting tweet: "I can understand why the idea of meaningful oversight terrifies the President. Several of his close associates are going to jail, others await trial, and criminal investigations continue."

He also gave Trump something else to worry about, as his committee voted to hand over transcripts from its Russia investigation to Mueller.

Interviewees concerned include the President's son Donald Trump Jr.; his son-in-law, Jared Kushner; and senior campaign aides Corey Lewandowski, Steve Bannon and Hope Hicks.

There is no indication that any of those witnesses is in legal trouble. But in his indictment of Trump's former personal lawyer Michael Cohen, Mueller has already shown no tolerance for lying to Congress.

Trump's new reality

Trump probably had few illusions that his decision to mention the investigation in the State of the Union -- a step some political commentators had advised against -- would intimidate Democrats.

It was a revealing comment nonetheless, since it offered a characteristic glimpse into the mindset of a President who chafes at constitutional restraints and appears to reject the notion of congressional oversight.

Democrats are using just that point to push back at Trump's warnings.

"I take it as another example of him not understanding how the democratic process works," Rep. Karen Bass, a California Democrat, told CNN's Kate Bolduan.

"He said, 'It just doesn't work that way.' No, actually it does. There are three coequal branches of government ... this is the new reality that he's going to have to get used to."

Trump's blasts may also indicate that he is beginning to comprehend the implications of the Democratic capture of the House -- and its investigative powers -- in the midterm elections in November.

They represented the first time before a mass audience that the President had voiced the building blocks of a political argument that could see service during any Democratic impeachment effort or in the 2020 election.

He was effectively arguing that the soaring US economic expansion and high job-creation numbers under his watch would be put at risk by any possibility that he could be forced from office or is unable to perform his duties without distractions.

Trump's warning may also be an early attempt to discredit the congressional investigations -- and to offer a soundbite for his conservative media friends -- in the same way he pounded away at the credibility of the Mueller probe. His efforts may not have dented the special counsel's reputation among the wider public but they may have discounted Mueller's eventual findings among the people Trump cares about most: his base.

Trump's defense

Trump may also be playing an internal Washington game with his State of the Union calls for unity and compromise already looking threadbare.

By using the word "legislation" he implied that Democratic attempts to use their House majority to create a record of new laws to lay before voters next year could founder in the fevered atmosphere of a scandal-obsessed Washington. After all, the President is needed to sign bills into law.

He could also use the conceit to explain to his supporters and more moderate voters why the second half of his own term is unlikely to yield any big legislative victories. Trump is likely to tell voters in 2020 that Democrats were so obsessed with destroying him they forgot America's business.

The argument would take on more potency should Mueller fail to find any wrongdoing by the President, even though his probe has already won convictions and guilty pleas against a number of Trump's former associates.

Yet Democrats can counter that they owe a duty to the public, and to their own constitutional role, to find out whether any of Trump's business interests and entanglements are influencing him in a way that endangers national security and therefore merits impeachment.

And their investigative offensive, which will involve a number of committees, will peel back the Trump administration layer by layer and offer exhibits for their own case in 2020 that the current White House is corrupt and unfit and that Trump should not be returned to power for a second term.