Let's say, for instance, you as a man were to be sued for child
Generally speaking, the cause of action is to establish paternity, which can be brought by a variety of parties, and child support merely follows from the existence of paternity. Often a possible father of a married woman does not have standing to contest her husband's presumed paternity and only the husband, the mother, or a representative of the child can bring the action.
What would be the outcome?
The question deeply misunderstands what a birth certificate means in the context of a paternity case.
Most U.S. states have paternity laws that are structured such that paternity is presumed under certain circumstances. Being named as a father on a birth certificate is one such presumption (and whether it is rebuttable or not depends in part, in most jurisdictions, upon whether the father named consented to be named on the birth certificate). See, e.g., this Law.SE question (analyzing California law and linking to answers under New York State and Minnesota law), this Law.SE question (further examining California law and DNA testing) and this Law.SE question (addressing related matters regarding non-marital child bearing).
Some of those circumstances can be rebutted with contrary evidence, and some cannot be rebutted (e.g. paternity by adoption or consent cannot be rebutted). Special rules apply in cases involving sperm and/or egg donation that are an evolving area of law and vary considerably from state to state.
Sometimes the grounds for presuming paternity include more than one possible father, sometimes they include none, sometimes they include one possible father. It is also possible in some states under some circumstances to have more than one legal father (e.g. if there is an adoption because a father was presumed dead and turns out to have not died, or in the context of a same sex marriage).
There is also a statute of limitations after which a presumption of paternity cannot be rebutted if it exists for child support purposes, which varies from state to state, typically one to five years, and typically a child can seek a paternity determination if there is no presumed or legally adjudicated father at any time until a few years after turning age eighteen. This statute of limitations doesn't always apply for other purposes (e.g. inheritance).
In circumstances where the presumption of paternity is rebuttable, or where there is no presumption of paternity but there is a showing (e.g. by a sworn statement of the mother) that the person sued for paternity is a possible parent, usually a DNA test is administered to the possible father and the child, and that resolves the issue conclusively (except in vanishingly rare freak cases, for example, of identical twins who both had sex with the mother, or alleged wrongdoing in the conduct of the DNA test).
If paternity is established, then child support is awarded based upon custody arrangements and the incomes of the parties and extraordinary expenses and in certain extraordinary cases, based upon other factors (e.g. a father or mother with great wealth but little income).
After a legal adjudication of paternity, the birth certificate is also amended, if necessary.