Sure, it would... if both the House and Senate decide its impeachable. Impeachment is governed by somewhat subjective criteria, but it's in the Constitution. Without going into detail, if enough representatives in both houses really don't like the current president, they can remove him via impeachment. Proof of a crime really isn't necessary.
The House can vote for articles of impeachment... a sort-of indictment, only requiring a simple majority to pass.
The Senate holds the actual trial. If 2/3 of the Senators vote for impeachment, the president is out.
In today's situation, it is quite possible that the articles of impeachment might pass the House. Getting a 2/3 majority in the Senate to convict is unlikely, if for no other reason that removing any president is a highly destabilizing event in itself.
Looking back to the last unscheduled presidential departure, Nixon, the Democrats won big in 1976. But, the generally uncertain nature of a president being forced out of office, combined with Carter's massive tax increases, had investors fleeing the stock market for bonds and gold, which pushed the country into a deep recession in the late 1970's, with unemployment higher than 2008-2009.
For which the Democrats were held responsible in the 1980 election... leading to Republicans dominating the federal government for the next 12 years.
Impeachment should not be taken lightly. The unintended consequences can be severe.
If Joe Biden wasn't a 2020 candidate, there is a possibility that the investigation into his and his son's dealings with Ukraine would have moved forward.
Even discounting the right wing exaggerations, a US VP's son with no management experience at all and a somewhat irresponsible past including getting booted from the Army for drug use, suddenly getting a $600k-900k a year job as a director of a foreign energy company... looks like an attempt by that company or country to purchase Dad's influence.
Dad using the power of the VP's office to exert influence beneficial to his son, and bragging about it on video, does nothing to paint that situation in a more ethical light.