This is a complete guess on my part, because I don't have enough time to test it this morning. But try using a flush space between each of the characters on the path. A flush space is designed to adjust so that the space between each character is equal even if you add or subtract characters (or change the width of the text box. I think if you set your text to full justification, and used flush spaces between the letters you might be able to achieve this effect. Here's a link to the article where I first heard about flush spaces.
However, I don't know if full justification OR flush spaces will work with text on a path, so it will take experimentation on your part.
You add a flush space by going to the Type menu, and choosing Insert White Space.
EDITED TO ADD: Well there seems to be differing opinions about whether or not this works, so I decided to try it out for myself.
First I drew a circle, and then placed text on it using the text on a path tool. I typed the letter A, then inserted a flush space, then the letter B, inserted a flush space, etc. Once I had all of the letters typed in, I set the text for full justification. And this is what I got:
Obviously, there is not enough space between the A and the Z in this version, so I dragged the text boundary lines (those vertical lines with the boxes that you see between the A and Z) apart a bit. I didn't worry about making this distance approximate the space between each letter, but it would be easy enough to do just by adjusting the spacing until things looked correctly. Here's how it looks after I separated the text boundary guides:
You can see that the spacing between each letter has been slightly reduced so that the letters are still fully justified.
Then I used the bounding box corner and reduced the size of the circle. The original circle was approximately 30 pixels x 30 pixels; the new circle is about 25p x 25p. Resizing the circle in this way doesn't impact the font size, which is 30 points in all three images. Here is the smaller circle:
You can see that the letters are closer together, and the text boundaries are further apart--you would have to readjust these boundaries to make the spacing seem more even all the way around, but again that can be done by eye, and should be fairly simple.
This will not work if you try and use tabs, or if your text is in tables. But I inserted a vertical line glyph on either side of the first few letters on my smaller circle and you can start seeing how this can be made to resemble text in separate boxes--experiment with hairspaces and thin spaces between the lines and their accompanying letters to see what you can accomplish.
It may not look exactly like what you originally showed in your picture, but I think it has the potential to come pretty close to what you were trying to accomplish, with a minimum of manual copying and adjusting.