Canada has hundreds of rivers suitable for your trip in most ways.
Canada, especially Northern Canada, is famous for its waterways, and for Canadians navigating them in one way or another. Of course the most normal way to do it is by canoe or kayak, but there is as far as I know nothing to stop you using a raft if you feel like it. In fact, depending on your definition of 'raft', many groups organize trips on them.
In Northern Canada (by which I mean anything more than about 100km from the US border!) you will have no trouble finding rivers that are navigable, allow rafts, have no commercial traffic,and take a trip of a week or more. Finding a town will be a bit more tricky, although most will have settlements at least somewhere along them. Those settlements will certainly give you a different cultural experience, even if you are born and bred in Canada. If you want a real adventure, go to the far north - any of the rivers emptying into the Arctic. That's all of Yukon, NWT, Nunavut, and the north of the big provinces.
I'm not sure how you were planning on getting your raft - most outfitters are geared for canoes and kayaks, which are going to be better in the whitewater you are likely to encounter at some point, and in the portaging you will necessarily have to do. You might persuade them to outfit you with an inflatable raft. I suspect this is going to be your major difficulty, wherever you go. I assume you were not thinking of cutting down trees and making one.
If I might make some suggestions of trips I'm personally aware of:
- Grand River (Ontario) Easily accessible, rafts for rent, plenty of towns on route, but probably not quite long enough to last a week. Also a bit short of camping spots. I know people who have done two and three day canoe trips.
- French River, Ontario. Plenty of outfitters and a few small towns. Suitable for a longer trip.
- Moose River, Ontario. Much more wild but relatively accessible (by which I mean you can get in and out by train instead of plane). Only one significant settlement. Outfitters will organize the trip. Plenty long enough to last a week. Also not cheap.
P.S. If neither 'raft' nor 'river' are essential to the experience, but just a water-bourne adventure - or you feel the need for a 'warm-up' trip - I strongly recommend canoeing any of Canada's national or provincial wilderness parks. Algonquin or Killarney are ideal for Ontario, but there are others in every province. it's surprising how little you have to travel to get a real wilderness feel.
P.P.S A raft would absolutely not be the ideal choice for this trip. They are heavy, hard to portage (a 'portage' is where you have to carry your boat because the river is too shallow or too rocky), less controllable, vulnerable to rocks, less good at carrying loads, slower, harder to refloat or right, etc. A canoe will easily carry 2 people and the gear you need for a week's trip. There is a reason canoes have been watercraft of choice in this region for the last five hundred years. And the difference in the experience of floating down a river in a canoe and floating down in a raft isn't that much.