Not for the electoral college. For that the House of Representatives picks the President and the Senate picks the VP. You should also note that in the House, normal voting rules don't apply: each state receives one vote, the reps from each state will need to caucus together to determine which candidate receives that vote.
It is pretty crazy. The equal-by-state House vote that you mention is particularly mind bending, an odd exception to population-based representation in the House. Furthermore, the District of Columbia would not have a vote in the tiebreaker, despite having 3 votes in the Electoral College.
