Of course, proof of absence is a very hard thing to achieve, but I'm going to argue that the US Army at least absolutely did not organize military brothels in Western Europe.
My main source is What Soldiers Do: Sex and the American GI in World War II France by Mary Louise Roberts. This book is somewhat unfavorably reviewed on Amazon by veterans who state that the book misrepresents the position of the Army (and of individual soldiers) at the time by suggesting that the Army actually enticed soldiers with promises of easy sexual relations. These veterans argue that the Army was actually discouraging them to have any sexual relations. Seeing that the main criticism leveled at the source is that it might paint the image of an Army to lenient on sexual mores, I think it safe to use it to establish my point.
According to Roberts, the US Army was very forceful in not condoning brothels in Western Europe as that would have in all likelihood seriously damaged the glorious image of virtuous fighters US soldiers had to project for the home population to remain in favor of the War. Besides, mass prostitution spreads venereal diseases and a well-organized army wants healthy soldiers, so that the official policy of the US Army was to warn against prostitution and to advise chastity.
It turns out that this health concern was not ill-founded, as 1) the use of the (then perfectly legal and institutionalized) French brothel system, 2) occasional prostitution and even 3) rapes all seem to have been quite massive in late 1944, early 1945 ; apparently especially in north-western France, with the expected health (and social) consequences. At least the first 2 were probably to be expected with the massive influx of relatively rich Allied soldiers meeting an impoverished French population who had suffered 4/5 years of occupation. This elicited official protestations from French officials, most notably the mayor of Le Havre, Pierre Voisin. He pleaded for the institution of an official military run brothel in the hope that this would help control 2) and 3), which, he described, was jeopardizing the already very fragile reconstruction of normal society in a city which had just been freed from a difficult occupation and had been heavily bombarded by the Allies. The Army adamantly refused.
Seeing that it refused a direct request from an official to do so, I thus think it very unlikely that the US Army ever organized any military brothel in Western Europe during WW2.
Not limited to Nazi-Germany and Imperial Japan the following belligerents are implicated in providing women for sex to military authorities, Indonesia, Hong Kong, Singapore, the Philippines, and Thailand. A series of searches revealed an informative article. A snippet of this credibly sourced article is provide below.
During the Second World War, approximately 3,500 Australian military nurses served in combat regions throughout the world. The vast majority were enlisted in the Australian Army Nursing Service (AANS), but after the Japanese advance and the fall of Hong Kong (December 1941) and Singapore (February 1942), a significant number of these nurses spent three-and-a-half years as POWs in Indonesia, Hong Kong, Japan and the Philippines.1 To date, considerable research has been undertaken on POW experiences in Hong Kong, Singapore, Thailand and Japan, albeit primarily focused on the testimonies of men and civilian women.2 This body of research utilises various methodologies, from Yuki Tanaka and Kei Ushimura’s efforts to reconcile Japanese war crimes with the corruption of the Bushido ethic and sexual violence in contemporary Japanese society, to Christina Twomey’s work on the imprisonment and repatriation of Dutch, Dutch–Eurasian and Australian civilian women and children. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3143890/
All the major combatants of the war established brothels of one type or another. The US army organized brothels by division. So, every division had its own brothel which moved as the division moved. Although they were "privately" run, the army kept tight control over them. The French, in particular, were extremely offended by these brothels. Unlike the German brothels they were completely indiscrete, being gigantic warehouse-type affairs simply plopped down in the largest roadways. For example, after the invasion of Normandy we made Cherbourg our port. When you walked or drove into Cherbourg on the main road from the south the first thing you saw were two gigantic brothels, one on either side of the road. One was for white troops, the other for "negro" [sic] troops (as they were called at the time). The US Army practices in setting up, supplying and maintaining these brothels were extremely offensive to French sensibilities and significantly damaged US-French relations.
see "The Price of Discretion: Prostitution, Venereal Disease, and the American Military in France, 1944–1946." American Historical Review (2010) by Mary Louise Roberts.
My great uncle Edwin Sayre passed away recently, and I believe was a real war hero, and involved in the US military up until the Vietnam War while advising for the President's Security Council. One of his many memorable stories from World War II that Uncle Ed told was being put in charge of a German brothel right after the war ended, which was for American troops. I always got a chuckle when he said that the first thing he did was double the price of girls, otherwise that was way too much temptation for a man to handle.
Just to add to Tyler Durden's answer which gave examples of the US military regulating / enabling brothels in France during WWII. I'm familiar with similar activities by the US military in Hawaii during WWII, where the Navy established price controls and red light districts for the practice.
Prostitution in Hawaii
The "bawdy houses" soon set up in Chinatown. An era of unofficial regulation followed, which was endorsed by the US military. There were struggles between the police and military, neither wanting to be seen as being in charge of the red-light district.
During the early 1940s the area served 250,000 men per month, at a fixed fee of $3. On 28 August 1942, the prostitutes of Hotel Street (the main street of the red-light district) went on strike for better conditions and the right to work away from the brothels. The strike lasted 22 days.
After the attack on Pearl Harbor, people were fearful of further bombings. People were keen to sell their property and O'Hara brought up several properties which subsequently made her very wealthy. Immediately after the attack, many of the brothels were turned over to quarters for the wounded. With the brothels occupied and the authorities distracted, the prostitutes started working outside the district. O'Hara started working from the luxury Moana Hotel with two other women. Following complaints about noise, O'Hara was arrested. As the military had now taken control of the red-light district and prostitution, she was tried by a military court and sentenced to 6 months.
On her release she returned to her brothel. which was on Hotel Street. By this time the military had set a fixed price for prostitute's services. O'Hara is credited with inventing the "bull pen" system where a single prostitute would work three rooms in rotation: In one room a man would be undressing, in a second room the prostitute would be having sex, and in the third room the man would be dressing. With price controls circumventing the laws of supply and demand, O'Hara's system sped up the process and allowed each prostitute to see many more 'johns' every day.