I know, a dumb question but I'm not getting any smarter until I ask. My cruiser's tire says 26 x 2.125. Replacement tubes are 26 x 1.75 - 2.25 and when I checked on new tires they said 26 x 1.75 - 2.25. So does the rim alone determine if it's going to be 1.75 or 2.125?

The rim determines the diameter.

You'll get a better understanding of what's what if you abandon the traditional 26 x x.xx sizing and go the the standardized metric, aka ETRTO size system. Those are the smaller numbers in this format -- xx-xxx.

The first two numbers are the nominal width (actual width varies slightly), and the second are the standardized rim diameter where the tire seats.

So 26 x 1.75 would be 45-559. All 669 tires will fit the same rim, and you have a decent amount of latitude about width, assuming the frame or fork are wide enough to clear them. In theory, any 559 tire will fir the rim, but if you go disproportionately wide or narrow in relation to the rim, the performance will be less than optimal. Here's a rough guideline for matching tire and rim widths, along with general tire info you might find useful.

Rims of a given width can take tires with a range of widths. I don't know how wide your particular rims are, but if the 2.125" tires work well for you, you've probably got a little bit of wiggle room to go narrower or wider if you prefer. Maybe down to 26" x 1.75" or up to 26" x 2.25" ...so long as there's enough clearance so a larger tire won't rub your frame or fork.

And tubes, since they're elastic, can fit into a range of tire widths. Once you know the exact tire size you'll be using, choose a tube whose labeled range of widths includes your particular tires' width. For example, if you're running 26" x 2.125" tires, a tube that's labeled 26" x 1.95-2.25" will work just fine.

a couple of my favorite references for figuring out if my rim is wide enough for the tire I want to use:

Tire Sizing Systems

https://www.schwalbetires.com/tech_info/size_markings

Page 10 of this PDF for rim width to tire diameter: https://www.schwalbe.com/en/cataloge...fo-2015_GB.pdf

-- is your question how fat the tire is? The diameter of "fatness"? Or how BIG the WHEEL is?
As mentioned in the first response, there is the WIDTH of the tire, FATNESS.
Then there's the SIZE of the wheel, how tall the wheel is. The tire you can fit on the rim is determined by the Bead Seat Diameter, and how fat of a tire you can fit on your bike depends on bike clearances. And of course the FATNESS affects how TALL the wheel is too.
What's really confusing is you can get an old "27 inch" road bike wheel from the 70s and 80s. The RIM is bigger than a 700C wheel you can get then and now.
BUT when you buy a "29er", the BSD is the SAME as the 700C wheel, but it's called a bigger size than the old 27in wheel! Which actually has a "taller" rim..... ugh.
And make sure you know the difference between your 26in MTB wheel and your 26in cruiser wheel from the '60s!

Any way, two cents worth. Cheers!

Would never go bigger but thought about putting 1.75 on it since I'm mostly on pavement. But the replacement tire is variable (1.75 to 2.25).

That doesn't seem right. What kind of tires are you looking at? Maybe they're available in multiple sizes and you're just reading the range they're available in?

You may be right Sky. Went back and found a box of 1.75 blackwalls.

No, went back and found the tire. It's a Bell and plainly stated 26 x 1.75 - 2.25. Another Bell did say 26 x 1.75.

It looks like the box is labeled to show that the tire could be used to replace a range of existing tire sizes. Very odd if the tire itself would be labeled like this... some product review comments say that it is stamped as 1.75.