Below is a scenario. It is not a real life situation, but more a hypothetical so I can get clarification of a specific of networking behavior.
Say we have a single PC directly connected via ethernet to a router on the private side. Public side of the the router is connected the the internet via whatever means. Router has NAT enabled, and a single port forward rule. Port forward rule is configured to send incoming traffic from public side to single PC on private side. (lets say to allow TCP pings) PC on private side does not have a default gateway set. Client on public side sends a TCP ping to public interface of above router.
Will the public client get a response ? Does the NAT modify the source IP in the incoming public packet to it's own IP before sending it to the private PC, so that the internal PC knows where to send the response ? Or does standard NAT implementation not modify the source IP of the packet, and therefore the internal PC must use a default gateway ?