Which sentence is grammatically correct?

The selected sentence "He felt bad about the broken window" is grammatically correct because it uses "bad" as an adjective, which appropriately describes the subject's emotional state concerning a situation. In this context, "bad" functions to express a feeling or moral discomfort.

Using "bad" in this way is standard in English when describing feelings. It aligns with the rules of English grammar whereby "bad" is the correct form to convey an emotional state. This contrasts with instances where one might use "badly," which typically describes how an action is performed, such as in physical skills or activities.

The other statements either misuse "bad" as an adjective or confuse its function in the context. This grammatical understanding highlights the importance of knowing how to apply adjectives and adverbs correctly based on the intent of the sentence.