I have just recently watch The Knick about a Hospital in 1900/1901 and in it all Doctors, Nurses, and EMT’s (more like actual ambulance drivers in this time rather than medical personnel but they were handling some pretty nasty things) and no one had gloves, mask, or any other PPE (personal protective equipment) other than Hand Washing. Just a EMT student who is curious and loves both history and medical knowledge.
No one quite knows who first introduced rubber gloves into surgery. Galvanized rubber and subsequently gloves were developed in the middle of the 19th century. In surgery, this was the time of Lister, Koch, and Pasteur (that is, the acceptable of sterilization, then germ theory). As such, by the 1870s in Europe (and by the 1890s in the US, where the surgical culture lagged in accepting sterilization and germ theory) it had become standard to spray carbolic acid throughout the operating theater. Carbolic acid sterilization was discovered actually even before germ theory was widely accepted, and it was sprayed everywhere -- including over the patient and into surgical wounds. This naturally was very toxic to surgeons' and assistants' hands, which is why rubber gloves started to be used. Again, it's unclear who was the first to do it. Rutgow in Archives of Surgery suggests it was a Dr. Thomas of NYC in 1878. While he may be the first American, certainly there were French and German surgeons using these gloves earlier in the 1870s to protect themselves and their staff from carbolic acid.
But this is not how they became widespread (in the US). The story is quite famous. William Halsted of Hopkins (one of the most famous surgeons in US history, and notoriously a cocaine-then-morphine addict) introduced them pretty much accidentally. He used mercuric chloride rather than carbolic acid for sterilization. His scrub nurse developed a horrific dermatitis to the substance, and he devised a new type of rubber glove for her (he would later go on to marry her). However, he noticed that cases she was on had a lower infection rate, and subsequently had his entire team use rubber gloves. This was part of the so-called "Halsted technique" -- the use of rubber gloves (and also very long surgeries where he was able to do increasingly complex procedures), and it was a major piece of surgical culture that was exported from Hopkins to the rest of North America. At the same time, in Germany, Kronig had run experiments at the same time on animals showing the superiority of rubber gloves to washing hands.
By roughly 1900, a vanguard of surgeons in the US and Canada were using rubber gloves in the OR; I honestly don't know as much about surgical literature in general, let alone what was happening in Europe. In any event, surgeons are generally quite resistant to change, and it wasn't until the 1920s that rubber (and then latex) gloves had become standard in the OR.
Hope that answers your question! Here is a nice article about the Halsted's and has a photo of a preserved glove https://whyy.org/segments/how-a-romantic-gesture-helped-bring-surgical-gloves-to-american-operating-rooms/
Also, if you like medical history, I host a monthly podcast with the American College of Physician's called Bedside Rounds. You might like it! https://itunes.apple.com/us/podcast/bedside-rounds/id919579524?mt=2 or www.bedside-rounds.org
I'm also currently reading Bliss' biography of Osler -- fascinating look at Hopkins during this amazingly formative time in American medicine!
EDIT: don't write reddit posts on phones