When presidents believe congressional actions are unconstitutional, they often refuse to cooperate. Congress may threaten contempt and impeachment, but it cannot make the president comply. Our brows furrow as we try to sort out the legal ramifications of all this. But in the main, the fallout is not a legal dispute; it is a political contest.
The Constitution is designed to promote both cooperation and competition between the political branches. Often, the judiciary stays out of these duels, prudently reasoning that the Framers endowed the executive and the legislature with powerful tools to confront each other.
The political damage sustained by the side that appears to be acting unreasonably becomes a powerful incentive to compromise. Public opinion is the court that matters here.
The president is correct that the Constitution gives the impeachment power to the House of Representatives. The power is not given to the speaker of the House or to a cabal of partisan committee chairmen. It belongs to the House as an institution.
The House acts as an institution by voting. On the matter of conducting an impeachment inquiry, the House has not voted.