Actually, if your western monk concept goes on some kind of pilgrimage (let's say for example, you make it a personal goal to visit all the shrines on the server) you would still be a western type monk, but at the same time you would be traveling all the time much like an adventurer would. In most cases those pilgrims will probably not be carrying any weaponry (or any other possessions for that matter) with them. So if they were somehow forced to fight, hand-to-hand combat or using their walking stick to defend themselves sound like very viable options and I can imagine them learning some basic self-defense before they venture out into the world. They will not be trained to the extend of a true martial artist, but enough to fend off a cutthroat or highwayman surely.
The Elder Scrolls: Morrowind, for example, has a traveling monk as a pre-set choice. Their monks bridge the gap between eastern and western culture quite well.
And I'm not sure, but I think Friar Tuck was (in some stories at least) a traveling monk on a pilgrimage of sorts. Although his martial prowess is definitely not asian.
That said, the cleric class will probably still be a better match for your concept. Or if you do end up rolling with the monk class, you should make sure to at least avoid the more kung fu inspired abilities. Which puts you at such a huge disadvantage, in regards to combat mechanics, that in the end you would have been better off picking a cleric or fighter class anyway.
The one upside would be the ability to use monk only gear, which does have very fitting stats for an unarmed fighter and not all of it is necessarily restricted to the eastern culture. (Praying beads come to mind.)
So in conclusion, if you plan to roll a friar or pilgrim type of monk, you are better off going with a different class than monk.
(Unless you are willing to seriously restrict your combat effectiveness and refrain from using a lot of items, feats and abilities.)