One word for playing on the violin with a bow

bowing

How to Use Exquisite Bowing Techniques on a Violin. http://www.wikihow.com/Use-Exquisite-Bowing-Techniques-on-a-Violin

There are several specific adjectives phrases to describe the exact manner of bowing, but the general word is arco. https://en.wiktionary.org/wiki/arco#English

Here is a good introductory glossary of specific techniques: http://beststudentviolins.com/terms.pdf

<digression>

Arco is the Italian word for bow, so the word for bowing is essentially "bowing", and when speaking in English--especially in informal situations--it would be acceptable to simply use the word bowing. Like many musical terms however, the traditional Italian version is preferred by those-in-the-know.

</digression>

<unwarranted pedantry>

While one may say generally that a guitar is played by strumming, strumming is only a category of articulation (like arco is a category of articulation). Specifically, strumming is a continuous stroke by finger or pick across several strings; if only one string is articulated we categorise that a pluck instead. Plucking can be further described: up strokes and down strokes, free strokes or rest strokes, finger strokes or picked strokes, etc... In other words it is inaccurate (or at least neither specific nor rigorous) to say that guitar is played only by strumming. A variety of articulations exist on guitar and strumming is a class of articulation.

Likewise there are a variety of arco articulations: up stroke and down stroke, sul tasto or sul pontecillo, detache or legato, etc...

On viols (the violin's family) strumming is technically possible, but incredibly rare. Plucking (one string) is what you are thinking of. In orchestras this is usually called pizzicato (abbreviated pizz) because that is the Italian word for "plucked".

In practice arco is often used as a sort of antonym to pizz. So a violinist might see a part of her music marked pizz where she should pluck and a part marked arco where she should use her bow again.

</unwarranted pedantry>

As reward for reading this rambling wall of text, here's a pretty sweet cover of Jimmy Page's solo from Dazed and Confused, which is played with a bow (arco) on a guitar: https://www.youtube.com/watch?v=7R6yUyUPd7o

I think you are referring to fiddling. The "normal" way of playing a violin is to fiddle it.

fid·dle (fĭd′l) n. 1. A violin, especially one used to play folk or country music.

v.intr. 1. To play a fiddle.

v.tr. 1. To play (a tune) on a fiddle.

To stress the fact that one is using the bow (e.g. as opposed to strumming or plucking the strings in a pizzicato), then you would use the term bowing.

But for the subtler meanings I'd ask http://music.stackexchange.com .

Just for completeness' sake (there are already better answers): Playing.

Generally speaking, if you are writing a story or something and say someone was playing the violin, the assumption is that a bow is involved. It will provide the right mental image, and as the most familiar and common term, is probably what you want in almost all cases.

But, because you are asking this question, and used the word "playing" in the question, I'm assuming this is not one of those common cases.