The columns must be resized individually. However, the column width will persist within the same browser until you change the size again.
I started to write this on January 25, but didn't have time to finish it. Erin's technically correct but incomplete message prompted me to recover the message and continue.
This functionality is not currently built into the gradebook.
There is a non-API call that is made whenever the column width is adjusted, but it only supports a single column at a time. That means that you would need to call it for every column in the gradebook.
There are different types of columns in the gradebook, each identified by a CSS class: primary-column (student info), custom_columns columns (notes), assignment columns (scores), assignment_group columns (totals for assignment groups), and a total_grade column (overall grade).
That means that one could write a script that would get all of the IDs and then make the API call to save the column width for all of the columns. I had an idea about how to do this with a user script, but generally speaking, I think it's a bad idea to make every column the same width. It would be better to make certain types of columns the same width rather than a blanket one-size fits all. That would mean you would need to detect what type of column you were in based on where you clicked and then resize just those types of columns.
Overall, modifying the CSS is probably a better way to go and that can be done by the local user by a user script as well. There may be an issue with the column width overrides, but it may also provide the change you need without having to override them individually. I haven't dug into this enough to know.
What Erin wrote is true in as far as she goes, but it potentially gives a false picture of what is happening. The changes do remain within the same browser, but the widths are saved on the Canvas servers, not in the local browser. If I change the width in Chrome, it is reflected in Firefox and vice-versa. If I were to open up a brand new browser, it would reflect the widths seen in the last browser. The use of "same browser" may cause someone to infer that it is stored locally within the browser and not on the back-end and that removing cookies or the clearing the browser's local storage would reset the values.
I have noticed that column width also prevents you from opening the side panel 'grade detail tray'. Surely this is a bug, it can't have been designed intentionally to prevent you from opening it unless the column is a specific width.