Which two days of each year are there no major (baseball, football, hockey, and basketball) professional sporting events played?

The two days each year when no sporting events are scheduled are the day before and the day after the major-league All-Star Game. With the basketball and hockey playoffs already finished, and the football season a few months from getting underway, baseball is the only sport with games scheduled in July. As there are no baseball games scheduled before and after the All-Star Game, those two days are the only days each year without a professional game scheduled.