Can a father relinquish parental rights by notarized document in Michigan?
Absolutely, Not has to be through a court of law.
You can relinquish parental rights, but it does not absolve you of paying child support. When you relinquish your rights as a father, you are still a parent under the eyes of the law. You may also want to consider who's best being served by you removing yourself from your child's life just so as not to have to help raise him or her.
In all 50 states, you cannot relinquish parental rights "just because" unless a child is being placed for adoption. Otherwise, there would be an influx of people trying to get out of paying child support or meeting other parental obligations by signing over their rights. Under certain (and dire) circumstances, parental rights may be involuntarily terminated by the court.
No. To voluntarily relinquish parental rights in the US, you must be able to give a 'good cause' reason for it. One good reason would be in preparation for the child to be adopted. Trying to avoid responsibilities such as paying child support is not a good reason, the application would be dismissed.