Is it obligatory to inquire about the ingredients of food offered by Muslims, such as nutmeg and animal rennet, or is silence preferable to avoid embarrassment?

The default rule regarding cheese and foods offered by Muslims is permissibility and lawfulness. The same applies to food offered by disbelievers. There is no need to ask or inquire about such foods unless one knows or strongly suspects the presence of something forbidden, for the unknown is as the non-existent. The Prophet, peace and blessings be upon him, and his Companions ate from the food of the People of the Book without detailed inquiry. Therefore, if food is offered to you, and its default state is permissibility, you are not obligated to inquire further.