Admittedly, your father could have used any of several different varieties and serial ranges of M1903 rifles. But that one has far and away the highest probability of usage based on the information you provided.
You stated that your father served on a Destroyer in the Pacific. I assume that the Destroyer was newly manufactured and outfitted during the early days of WWII.
The Navy had many thousands of M1903 rifles, most leftover from WWI. But those rifles were already assigned to ships and installations that existed before the war. Any new ships coming on line had to be outfitted with newly-manufactured rifles because existing rifles were already committed elsewhere. And the primary source for newly-manufactured rifles during the early days of WWII was Remington Arms and that was the serial range. I have documentation from the President of Remington Arms quoting Army Ordnance officials that the Navy was clamoring for their rifles and urging expedited production.
I have seen perhaps more than a hundred Remingtons in the aforementioned serial range and they were definitely Navy issue.
Hope this helps. Happy Hunting!
John Beard
p.s.,
And I worked at the CMP.