You can, by changing InDesign's default Justifications parameters. However, you should not do nor want to do that.
Starting with replicating your result:
– this is 11/13 pt Times New Roman in a frame 60.5mm wide –, I found I had to insert multiple spaces in exactly those places in your "total mess". In modern typesetting, entering multiple spaces in running text is not only not necessary (in HTML, for instance, the default is any amount of spaces is equal to exactly 1), but as you can see it messes with the built-in justification algorithm.
By changing the allowed word justification limits to Minimum 100%, Desired 100%, and Maximum 100%, you can force InDesign to not change the space widths. Since apparently (and weirdly) you still want your text fully justified, you need to allow either intraword spacing differences (the letter spacing) or glyph scaling (or, as it is, a combination of the two). Just for laughs, I set the Letter Spacing to -10%, 0%, 10% and got this result:
This is what you asked for: all of the spaces have the same width, and the text is justified. Some of the interword spaces are larger than others, but that is because there are more spaces in those positions.
I could stop with this because it is what you asked, but I started out with telling you really should not do this. The proper way is:
- Do not have multiple spaces inside your running text. None. Not even after the full stop.
- Set letter spacing variance to an absolute minimum. InDesign's own defaults are
0%,0%,0%, that is, no variance at all.
- Set word spacing variance to reasonable values. InDesign's defaults are
80%,100%,133% and are perfectly alright for a wide range of fonts, sizes, and column widths. For small columns such as yours you may need to decrease the Minimum to about 75% and perhaps the Maximum to 150%, to prevent lots of words being hyphenated (then again, don't do this if you want your spacing to be more consistent).
- Rely on InDesign's highly advanced, patented Full Justification algorithm. Other than most typesetting software, InDesign distributes its spacing all over an entire paragraph. That means that every single line is justified the best way it can, considering all other lines as well (i.e., slightly changing the spacing on one line for the better will make at least one other line much worse).
With the Justification settings left at their defaults and with all of your double spaces removed, your snippet of text looks "just right" to me: