My wife has been struggling with paternity since we were both young she is now 41.her dad would not take a paternity test or accept her at all.now her dads parents took a dna test.my question is can the father be sued for abandonment?

No, you don't sue anyone for abandonment even if the child is young, you file for child support, custody, visits and more.

Abandonment in most states is a criminal charge for leaving a infant at a store or side of the road or home alone. In some states it is not paying court ordered child support.

The mother in this case just did not do what she was allowed or suppose to do.
She could have went to court and got a court order to force a DNA test and file for child support. ( blood test if the child is that old most likely)

So look and ask mom why she did not use the court 40 years ago and let the dad get away with it.

There is nothing you can do now.