What do you think would have happened if the Navy HAD gone into the Marshalls looking for her?
People would have questioned their sanity. On what grounds would the Navy have gone to search inhabited territory? What indication from the flight itself would have sent them that far away? How could they have ignored the
steadily-increasing signal strength which could only have come from the transmitter flying closer and closer to Howland?
Might history be different?
No. Nothing in the Marshall Islands would have pointed to the attack on Pearl Harbor 4+ years later. Yamamoto did not gain full approval for the attack until 1941.
Nothing in the Marshall Islands would have made the U.S. consider Japan more or less of a threat to peace. Japan conquered Manchuria in 1931, Jehol in 1933, inner Mongolia in 1936, and invaded China on 7 July 1937. That the Empire of Japan was militaristic and developed military installations in its territories would not have been a great shock to U.S. strategists. That the Empire might object to a U.S. naval incursion into its territories would not help waken the sleeping giant. That naval ships searching far away from where Amelia was reasonably understood to be might stumble across a military secret that would cause the U.S. to arm itself and go on the alert is not conceivable to me.
This is a "coulda, woulda, shoulda" argument. I'm imagining what could or would have happened. It is extremely difficult, if not logically impossible, to
prove this kind of negative: "U.S. naval ships searching the Marshalls in 1937 would not have changed the history of WW2." What would be required to make the case watertight is a complete set of aerial photographs of the whole of the Marshall islands that would enable us to see what the Navy could have seen after 2 July 1937. Alternatively, a set of maps showing the actual dispositions of Japanese forces in the Marshalls would be helpful, if one trusted the source that produced the maps. Only if there was something large and easily noticed from a distance would the naval ships have noticed it, become alarmed by it, and reported it to headquarters in such a way that the entire government would have been persuaded to invest in better defenses in 1937. I have never seen such an analysis of what was visible in the Marshalls; the absence of evidence is not necessarily evidence of absence, but I feel morally certain that the lack of such evidence
in this case does mean that
there was nothing to see in the Marshalls.
Your mileage may vary (YMMV).