House Speaker Nancy Pelosi (D-CA) speaks during a news conference on Capitol Hill on October 15, 2019 in Washington, DC. House Democrats will not hold a vote to authorize impeachment inquiry into President Donald Trump. Also pictured is House Intelligence Committee Chairman Rep. Adam Schiff (D-CA).

House Democrats will not hold a vote to authorize an impeachment inquiry into President Donald Trump as of now, House Speaker Nancy Pelosi said Tuesday, defying calls from the White House and Republicans to do so.

The decision to hold off on a formal vote comes amid growing pressure from Republican lawmakers and White House officials who have criticized House Democrats' efforts as an illegitimate attempt to undermine Trump's presidency.

But many members of Pelosi's caucus want to avoid the perception that the White House is dictating how the House, as part of a separate and equal branch of government, conducts itself, a congressional aide familiar with Democrats' discussions told CNBC.

Pelosi and House Intelligence Committee Chairman Adam Schiff, D-Calif., pushed back on critics who have accused Democrats of conducting the proceedings without transparency.

"We're not here to call bluffs," Pelosi said at a press conference Tuesday evening.

"This is not a game to us. This is deadly serious," she said.

The presser came just before Democratic presidential candidates were set to take the stage in the fourth debate of the 2020 primary.

Schiff said that "the Constitution is very clear: The House will have the sole power of impeachment."

"Republicans know it, but they don't want to discuss the president's conduct," Schiff added. "They'd rather discuss the process."

Pelosi has dismissed the calls for a vote as a "Republican talking point," and said in an interview earlier this month that "we're certainly not going to do it because of the president."

Republicans have seized on Pelosi's stance. Vice President Mike Pence, who had been sent a request for documents as part of the probe, on Tuesday said he would refuse to comply with the "self-proclaimed 'impeachment inquiry,'" according to a two-page letter to Democratic House committee leaders by his counsel.