Could you get a doctorate based solely on a thesis written in your free time?

Just write your paper, as long as the writing does not distract you from your undergrad studies. If the paper has merit, you may be able to present it at a colloquium of your university, at a workshop, or even at a conference. If you put some more work into it and are lucky, you might eventually be able to publish it in a journal. Certainly, your paper will open up more questions than it answers. Now you have a research trajectory that you can follow up and that may lead to a PhD thesis, based on but surely not limited to your first paper.

In other words: There is no reason why you should not start to work on your PhD topic already as an undergraduate, but it seems to me that you underestimate the time and work it takes to finish the PhD. And since PhDs require a lot of work, there are PhD programs, which facilitate the process and in which most PhDs are actually produced. In that sense, writing your PhD outside of the designated framework is a bit like digging a well with a spoon. It can be done, but it is not too efficient. On top of that (as others have said already), at many institutions, the written thesis is only one requirement among others to be awarded a title.

If my paper is credible, fruitful to my field and otherwise fulfilling the standards that any PhD thesis should

A research paper is not a thesis. Theses are usually expositions upon some field that the author has contributed to, and frequently contain extensive background that is typically omitted in academic literature. Indeed, many of my colleagues in industry have published research papers without having PhDs.

... would I be awarded a doctorate if I'd publish and wish for it?

In addition, you would need to find a university to endorse your work, which often requires collaborating with faculty.

Although, if your work is truly groundbreaking and you spend a lot of time hanging out at a particular campus, you might qualify for an honorary degree!

All this depends on the field, the country, etc. but in theory, if you already have a master's degree, it is possible to get a PhD based on previous (published) research. But typically not on a single paper. (Over)generalizing and assuming your research is genuinely interesting and meets all the (sometimes arbitrary) requirements of your field, a rough guideline is that you need material corresponding to at least three papers.

You would also need to write some intro/conclusion putting the research in context, make revisions based on your advisors' comments, find a jury and satisfy them. You cannot just show up with a paper and “wish” for a PhD, at least not from a real, reputable university. That's assuming you really manage to make your research publication-worthy (which means not only rigorous but also connected to recent literature and current problems in your field) and find a faculty willing to accommodate you as it's not the “normal” way to get a PhD.

Don't overlook the bit about finding an advisor: It's incredibly important. The PhD is really an apprenticeship under the supervision of a full professor. You don't ask a university or department for a doctorate, there is no process to submit a thesis and have it evaluated on your own, it's all driven by faculty members and you won't even be allowed to defend a thesis if it's not endorsed by one (he or she would also typically help you recruit a jury).

Realistically, a mediocre thesis can be validated if a professor puts his or her mind to it, asks friends to sit in the jury, etc. but even brilliant work is nothing if you don't find an interested academic to move the process along.

Beyond that, others have mentioned extra requirements but the thesis is really the most important thing. In my experience, in Europe (I got my PhD in the Netherlands but I know a little bit about France, Germany, or the UK), it's increasingly common to organise some mandatory courses for PhD candidates but the load is very light, a few short courses about soft skills or methodology with no exam, only pass/fail based on attendance. And there are ways around that if you have a good reason.

But to be perfectly honest, your question suggests you are not very familiar with academia so it seems highly unlikely you would succeed in getting a PhD without proper support. I am not sure why you want one but if it's important to you, it might be a better idea to simply enroll in a PhD program.