He felt embarrassed _____________________ his mistake in front of everyone.

The phrase "felt embarrassed" typically pairs with the preposition "of" when followed by an action, especially in the context of remorse or shame regarding that action. Therefore, "of admitting" correctly completes the sentence by expressing that his feelings of embarrassment stem from the act of admitting his mistake.

Using "to admit" suggests a focus on the action itself as a potential future event, which does not fit the context of already feeling embarrassed. "For admitting" introduces a different structure that could imply justification for his feelings rather than directly linking his embarrassment to the action. Lastly, "on admitting" would imply that the embarrassment occurred at the moment of admission, creating a slightly different nuance that may not align with the intended meaning of the sentence. Thus, "of admitting" effectively conveys the relationship between the feeling of embarrassment and acknowledging his mistake.