Is it permissible for me to ask a married man (who is accompanying his sick wife in America) to marry me with a mut'ah contract, knowing that I am also accompanying my sick husband in America, and after he refused my request to fast the White Days with me?

This woman is betraying her religion and her husband by claiming that temporary marriage is an act of worship to draw closer to that man. This is not a valid marriage; rather, it is zina (fornication/adultery), which the Messenger of Allah, peace and blessings be upon him, prohibited until the Day of Judgment. She must cut off her relationship with him immediately. The marriage of a married woman to someone other than her husband is not permissible by any religion or custom, and she must focus on her current husband.