United States Army

The United States Army is the primary ground force in the U.S. Military. The Army was one of Umbrella's bio-weapons customers,^[1] and operated its own special weapons program relating to the handling of weaponized viruses.^[2]

Umbrella and Raccoon City

Following his recent disagreements with Umbrella executives, William Birkin offered to defect to U.S. military authorities with the G-Virus. Knowing that possession of Birkin and the G-Virus would eliminate the need to purchase bio-weapons from Umbrella, the military accepted. The U.S. government decided to send a Special Forces unit to take Birkin into custody. However, Umbrella found out about the deal and sent the Umbrella Security Service to seize the virus from Birkin, while using spies within the U.S. government to cause the Special Forces to take the wrong underground lift. Rather than arriving at the underground laboratory, they arrived at the Incineration Disposal Plant P-12A. Meanwhile, the U.S.S. secured the laboratory and shot Birkin, who injected himself with the G-Virus and caused the t-Virus to leak into Raccoon City, sparking a major outbreak.

By September 25, it apparently appeared that most of the city was in ruins filled with zombies and other unseen creatures and the Raccoon Police Department (RPD) was unable to stop them due to the sheer numbers and underestimating the creatures' abilities, while the response to the fires in and around the buildings by the Raccoon City Fire Department (RFD) was severely limited by the zombie hordes. That same day, the State Governor deployed the Army National Guard to assist in the evacuation efforts of civilians in the city, and the President also committed large numbers of Army soldiers to bolster the National Guard. Due to the nature of the assignment, the Army and the National Guard were ordered not to enter the city and help the RPD and RFD, leaving the outnumbered police and fire personnel to fend against the zombies and evacuate civilians for themselves.

However, the Army did send two of its Special Forces units into the city, mainly to recover the G-virus. One unit was sent into the Raccoon General Hospital (which hosted an Umbrella bio-weapons and virology lab) and, based on Carlos Oliveira's observations, was wiped out by a pack of Hunter βs. Another unit was sent on a specialist operation into the Incineration Disposal Plant P-12A, where they killed and were killed by a number of T-103 Tyrants sent into the facility by Umbrella to stop them.^[3]

After Raccoon City

Once the Federal Bioterrorism Commission was disbanded, the United States decided to rely on its own Armed Forces to combat bio-terrorism in its territory. The Army's 75th Ranger Regiment primarily filled this role and was tasked with eliminating threats during the Harvardville Airport incident.

Further notes

In Resident Evil Outbreak, if a player decides to detonate the street infected with zombies, a scene will occur where the U.S. Army deploys an investigation unit into Raccoon City, presumably to identify the cause of the outbreak.
Also the character Mark Wilkins from the Outbreak series was a Lieutenant colonel of the U.S. Army during the Vietnam War.
It's possible the MPs escorting Billy to his execution are members of the Army, since they carry what is known as the Army Handgun (though it is never explained exactly how Billy obtained this weapon, this is the most logical assumption).
Billy's handgun is called the "Army Handgun" and is a modified Colt M1911. Examining the weapon explains that it is the standard handgun of the U.S. Army, which is inaccurate. While the U.S. Marine Corps still uses the Colt, the Army has switched to the Beretta 92FS (the weapon the Samurai Edge is built around). Likely this is either a translation error, or an attempt by Capcom to differentiate between both character's handguns.
The biographies for the original Resident Evil indicated that Albert Wesker, between 1991 and 1996, was a commissioned engineering officer in the United States Army, suggesting Wesker was a double agent.
In Resident Evil:Operation Raccoon City the USS (Umbrella Security Service.) Fight U.S. Army Spec-Ops there is also a dlc where you can play as the Spec-Ops Field Unit.
The U.S. Army Spec-Ops were sent in to evacuate civilians whilst the Spec-Ops Field Unit was sent in to find evidence as to why the outbreak happened.