The father will not have any rights to the child (or be obligated to support the child) until he obtains a paternity judgment from the court establishing his rights and obligations.
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If you get governmental assistance for the child, you will be asked to identify the father for child support purposes. The counties want the men responsible for these children paying support. I am unsure what happens if you fail to identify the father. If a paternity action is started, he in fact could request visitation right and might get them, but distance will play a factor.