This is actually two questions rolled into one because there is a definite legal difference between a spouse and someone you were married to but are now divorced from. Under law, married couples can act as a single entity while the ties between partners are dissolved when the marriage is terminated.
An ex-spouse should not have any financial ties remaining with you when the divorce is finalized. This includes separating loans, removing one partner from obligations faced by the couple as a whole, and removing unconnected spouses from your various insurance policies. When the relationship is over, the best action is to completely separate all ties, including your medical insurance coverage.
In contrast, your new spouse is entitled to share in everything with you. This does not mean that you are required to provide medical insurance for them, but it does mean that they can use your coverage if you choose to allow them to do so. You are not required to add that person to your coverage, but there is nothing standing in the way if you choose to do so.
In a situation where both partners bring their own medical insurance to the union, it is common for one partner's coverage to become the primary insurance while the other partner's policy is used as the secondary coverage. Which one is the primary coverage depends on the birthday of the two parties, with the first birth date, based on the day of birth rather than the year, serving as the primary coverage. For example, if your birthday is January 1 and your new spouse has a date of February 1, your coverage will be used as the primary and theirs will cover, hopefully, some of the gaps in your coverage.