I believe Tableau Public is pretty much like Tableau Server. It generally always runs the latest version of Tableau, so if you upload a 10.3 workbook, it will convert it to the latest version. I don't think you can force it to keep the older version.
What is limiting you from being able to use a newer version?
In an idea in the idea section someone asked to help with his workbook. He can't open any workbook newer than 10.3, issue is that I can't share workbooks within an idea but can't upload to public either, so the option is very limited. (Only option left is for that person to make a new discussion where I could share it).
I'd rather see the option for us to upload older versions to Public and share workbooks in the idea section.