US Military service records a are at the NARA www.archives.gov
At Ancestry.com they have the Disabled hospital record as well as a scanned image of his Veterans Pension card. The database "Civil War General Index to Pension Files" includes Spanish American War pensioners also. They also have a "U.S., Returns From Military Posts 1906-1916." that has him listed as discharged May 29, 1899. These may be the only records available. Our National Archives and Records Administration (NARA) had a fire in 1973 that burned about 80% of Army service mens records from about the SpAm War to early 1960's.
As for "Pacific Branch" refers to the West Coast of the U.S. area. The Government sometimes divides the US into areas or branches for command and logistics purposes. Pacific, South West, Mountain, Mid West, East. etc.
Did you see on the image scanned of Hospital record that they have him as enlisting July 16, 1898 in San Francisco, California. rank was Private in the Hospital Corps. Discharged May 29, 1899. Said his domestic occupation was a Nurse, lived in Jordan. Oregon, was single.
He filed for his Pension June 23, 1920 in California was in "Hospital Corp, USA" application number 1,434394 and certificate number 1189008.
According to the California Voter Registration database also on Ancestry. In 1922 he lived in Yountville, Napa Valley as a farmer and he was a member of the Republican Party. He 1926 he is still in Napa Valley he was a dishwasher in the Veteran's Home still a Republican. In 1926 he is a Druggist living back in Yountville, Napa County.
I also find him under the name as spelled Osean J Wallenburg in Return to Military Post database.