So, technically the Senate can't stop the House from moving forward with the impeachment process steps. It can, however, stop a president from being convicted and removed from office. In fact, in several instances throughout history, American presidents have been impeached by the House, but not convicted by the Senate — meaning they remained in office.
Remember this: Republicans control 53 seats in the Senate. Even if the Democratic-led House impeaches Trump, he will not be removed until 67 senators vote to convict him. Which, by a bit of back-of-the-envelope math, means that 20 GOP senators would have to side with all 47 Democrats (and independents who caucus with Democrats) in the Senate for Trump to actually be removed.
A simple majority of the House of Representatives (at least 218 votes) is required to impeach a U.S. President followed by a two-thirds majority vote in the Senate (at least 67 votes).