Is a divorce decreed by a non-Muslim judge in America considered religiously valid, given that the husband did not utter the words of divorce, and what is the ruling on the wife's second marriage after this divorce, whether only the marriage contract was concluded or consummation also occurred?

This divorce issued by the court is invalid, and the wife is still married to her husband. This woman's marriage to another man is invalid because she is already married, and this is an impediment to marriage, as stated in His saying, the Almighty: "And [prohibited to you are] all married women." If a woman dislikes her husband and fears she cannot fulfill his rights, she has the right to seek khul' (divorce at the wife's instance) or refer her case to a Sharia judge or Islamic centers. It is not permissible to resort to man-made laws except out of necessity. Residing in lands of disbelief is dangerous due to the trials a Muslim may face because of laws that might embolden a woman against her husband.