There's a lot of research/writing about this, but the thrust of the stuff I've read suggests that the effect of the Washington Post's work was exaggerated, a product of a romanticization of the situation in which the media created heroes out of Woodward and Bernstein, Wash Post, etc.... The movie made not long after Watergate "All the President's Men" (based on a famous book), no doubt romanticized/mythologized them further.

Here's a good source that debunks the popular myth about the Washington Post doing all of the work:

http://merrill.umd.edu/wp-content/uploads/2015/05/Watergate-Revisited.pdf

A useful passage:

The late Clark Mollenhoff, an investigative reporter who not only covered Watergate for the Des Moines Register but also at one point worked for Nixon, compiled a list of more than three dozen journalists besides Woodward and Bernstein who, he said, "made equally great contributions to the success of the Watergate probe." That undoubtedly overstates the case. But, says University of Virginia political scientist Larry Sabato, other reporters "got too little credit and the Washington Post got too much." According to Nixon White House counsel John W. Dean, as the scandal developed, the reporter "who does the most devastating pieces that strike awfully close to home was Sy Hersh," whom the New York Times assigned to the story. The Los Angeles Times also dug up scoops "of the same caliber of Woodward and Bernstein," says investigative reporter James Polk, then with the Washington Star, "but the L.A. Times wasn't read in Washington" as widely as the Post and therefore didn't have the same degree of influence.

There's also the suggestion by some that the news simply followed the case and it was really the slow-moving investigation of the criminal cases in the court system that eventually reached Nixon and that these legal problems were the major cause.

It seems clear that the Washington Post had some impact, at least in keeping the story alive and in front of the public so they knew what was happening (although the public didn't realize how serious it was getting until pretty late), but again -- the common depictions of their work tend to overshadow other important contributions and overstate their influence.

Here's the opinion of Stanley Kutler, widely believed to have written the most authoritative/definitive book about Watergate, "The Wars of Watergate."

"As more documentary materials are released," Kutler wrote, "the media's role in uncovering Watergate diminishes in scope and importance. Television and newspapers publicized the story and, perhaps, even encouraged more diligent investigation. But it is clear that as Watergate unfolded from 1972 to 1974, media revelations of crimes and political misdeeds repeated what was already known to properly constituted investigative authorities. In short, carefully timed leaks, not media investigations, provided the first news of Watergate."

The one aspect not discussed so far was the role of Judge Sirica and other judicial proceedings. Sirica was the judge in the original trial of the Watergate burglars and consistently complained about how there was more to the story, but was being hidden because the burglars kept their silence (they were being paid). It took James McCord's letter to the judge alleging a coverup to widen the story. The initial investigation was contained to the 5 burglars and Hunt & Liddy as "masterminds". This was part of the September 15th 1972 tape where John Dean is congratulated for keeping the investigation contained.

While the newspapers definitely helped keep the story active, they were regularly criticized by the White House for their use of unnamed sources. In addition, this was still a period where many trusted the government far more than they do now. It was essentially an "us vs. them" mentality, but the administration did a good job of keeping what was going on behind the scenes under wraps.

As for impeachment, I submit that there never would have been a serious discussion of it if Alexander Butterfield hadn't disclosed the taping system in his Senate testimony. Almost everything that followed built on the tapes, mostly because they became the proof necessary to show which side might be lying. Dean's testimony actually helped expose the tapes because for a variety of reasons he wondered to the Senate investigators privately whether Nixon might be taping (LBJ's taping was totally secret at this point). It was the tapes that eventually led to the Saturday Night Massacre, following which impeachment moved to the front burner.

Woodward's "The Last of the President's Men" tells Butterfield's story and Dean's memoir "Blind Ambition" is also useful to describe how the break-in and investigation was covered up.

Another good background book on Woodward and Bernstein is "Woodward and Bernstein: Life in the Shadow of Watergate", which covers much of the behind the scenes things, including how early Robert Redford was involved, before the book was even published.

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More simply, how much did public opinion about Nixon matter to the Congressional investigation and the eventual impeachment process, compared with the behind-the-scenes work of the FBI investigation? Was the Senate Watergate Committee formed solely because of the media coverage?