No, it is by no means universal. Frames are built with a maximum tire size in mind. Some bike include a maximum size in their specs, so you might check to see. Otherwise, it is easy enough to figure out. Since you have tires on the bike already, you can check the existing clearance and get a very good idea of what will fit.
Start by measuring the clearances. Here's where to look:
At the top of the fork – measure (or estimate) the clearance between the tread of the tire and the fork crown and the sides of the tire and the fork blades. One easy way to gauge the clearance is with a drill bit or Allen wrench.
At the chain stay bridge – here you want to look for clearance like you did at the fork (tread to bridge and side to stays) and you want to check how much room you have to install the wheel with the tire inflated. If you have vertical dropouts or track dropouts this isn't much of an issue, but with "old school" horizontal dropouts where the wheel moves forward in the frame to come out of the dropout, it can be the limiting factor (especially if you'd rather not have to deflate the tire to get the wheel out).
At the brake bridge – measure like the front fork. In my experience, this is not where the problems will lie, but double check just to be sure.
Once you know how much clearance you have you can get a pretty good idea of how a larger tire will fit. The "other" dimension on the tire is, roughly speaking, the diameter of the cross section on of the tire – imagine the wheel as a pizza, if you cut a slice and look at it from the side the tire is the outer crust (the thick round part), the 23 or 32 is the diameter of the "roll" of crust at the outer edge.
So, for you, to be able to go from a 23 mm tire to a 32 mm tire you need about 9 mm of extra radial clearance (clearance between the top of the tread and the fork crown and bridges) and about 4.5 mm of lateral clearance (between the tire walls and the fork and stays). I call this "extra" clearance because you want to maintain some additional space to allow for an untrue wheel and for junk that gets picked up by your wheels as you ride. On my Trek 620 I've got maybe 5 or 6 mm between the fork crown and my tire (they weren't thinking about fenders when they designed that bike), but 10-12 mm on the sides – aside from the fender issue that seems adequate (although it might not be if I was riding tires with a large open tread that tended to pick up rocks.
The final issue to keep in mind is that while the tires are spec'd as if they were round in cross section, many are not – the Schwalbe Marathon line for example is more oval than round, they account for this in their ISO sizes (they use the larger number, which on these tires is the radial dimension), but use the smaller size in the "700c" measurement. For example the 700x35c Marathon is a 37x622 in the ISO size. So to run this tire you'd need more "extra" radial clearance than side clearance.
