If the father of my child is married and he refuses to sign the birth certificate? When it is proven that my child is his, does his spouse have to pay child support if he isn't working?

No. The duty to pay child support is his duty, not hers. However if she desires to keep his property from being levied upon, to keep his wages from being garnished, to keep him from having his driver's license and passport taken from him, to keep him out of jail, etc. she may want to help him with his child support obligations.

No, the spouse is not required to pay anything, and normally just the income of the non custody parent is used, their spouse income is not considered in this. So if they are not working, the support order will be very low.