Nearly all colleges and universities in the United States and most other countries require a high school diploma or its equivalent (such as a GED) for admission. There are very few exceptions, and those exceptions are typically for exceptionally mature students with significant alternative qualifications or demonstrable academic achievement.
Therefore, the answer is almost all colleges accept a national high school diploma (or equivalent). However, simply having a diploma doesn't guarantee admission; colleges also consider GPA, standardized test scores (often), extracurricular activities, essays, and letters of recommendation.