Why is it that violins use simple pegs instead of geared tuners, like an upright bass?
Maybe you just never noticed them?
I would suspect that someone trying to sell a violin with a head like a classical guitar wouldn't be able to sell them into a very, very traditional market.
So... what did they do instead?
They made geared tuners that look exactly like regular tuning pegs - with the added bonus that you can swap them into any existing instrument with no modification.
At a glance you simply cannot tell the difference.
Regular peg? Nope, geared peg...
Pic from PegHeads
imo, Knilling has the best page to read about the reasoning behind the idea.
Some reasons why people don't use geared tuners on violins:
I haven't tested in personally, but there us usually some discussion on how using heavier material in metal geared tuners reduces some of the vibration transfer from the strings into the neck, causing some tone loss/difference. The wood to wood contact is supposedly better for tone.
The short scale of the strings and lower tension doesn't require geared tuning machines for ease of tuning the way longer or higher tension strings do (such as on a mandolin). The turning ratio of the violin strings is very low, especially when using gut based strings, making the 1:1 ratio of the peg more suitable for smooth tuning.
Geared tuners on violins aren't "traditional" looking, which is a consideration for some Orchestras, which may require a standard look. (in some cases some violin colors aren't allowed either, such as "blond" violins)
Geared tuners are more expensive. Manufacturers want to use the lowest cost solution, especially if it is already the accepted standard.
That being said, as stated in Tetsujin's answer, there are geared tuners available. There is even a manufacturer that makes violin sized Bass style geared tuners (I have a fiddler customer that prefers them, so I occasionally install a set on his new instruments).
I've found that the various designs of the geared peg style tuners have some issues. Over time the bushing or parts fail, and depending on the installation type replacing them or converting back to pegs can be an expensive job. In some models keeping the tension can become an issue as well. I'm seeing these problems on the older versions of the products as they come in for repair or on used instruments I'm setting up for sale, so newer versions may have some of the kinks worked out.
As already answered, the force of violin string does not require gears. If you asked due to the possibility of finer tuning granularity, there are fine tuners instead, which are much cheaper, but also do not convince all violinists: so they either resort to standard peg tuning or have them only for one or two strings.