Is it permissible for the questioner to avoid her husband's father's calls and refuse to speak with him, due to feeling hurt by his repeated and embarrassing questions about her pregnancy, especially with the threat of miscarriage?

We ask Allah to grant you healing and a safe delivery, and we advise you to make abundant supplications. Your father-in-law should not ask you questions that put you in an embarrassing situation. The principle is to avoid speaking about matters that cause embarrassment, unless there is a need for it and to the extent of that need, as stated in the Hadith: "Indeed, Allah is not shy of the truth. Do not approach women in their anuses." His intention may be good, but such directives are better left to a specialized female doctor. You are not obligated to answer his calls, nor are you required to obey your husband if he compels you to do so. However, to avoid discord, you can limit yourself to greetings and asking about his well-being, and politely deflect discussions about pregnancy by informing him that you are following up with a female doctor.