Congress can impeachment for illegality, yes. But technically, not just for unpopularity. Section 4 of Article II:
The President, Vice President and all civil officers of the United States, shall be removed from office on impeachment for, and conviction of, treason, bribery, or other high crimes and misdemeanors.
In the strict Constitutional sense, the President has to commit some kind of illegal act, or violate the law, in order to be removed from office. Bill Clinton's impeachment counts were perjury and obstruction of justice relating to his testimony under oath about his relationship with Monica Lewinski. In Andrew Johnson's case, Congress had passed a law preventing Cabinet members from being dismissed without Congressional approval. Which Pres. Johnson violated by firing Sec. of War Stanton. Congress wanted Johnson gone, and baited him by passing a law they knew he would violate. But the point is that some type of illegality--even a misdemeanor traffic violation-- is the only Constitutional basis for impeachment.
