The shorts answer is no. The __() and _e() functions are evaluated on the server side. So by the time the page is sent tot the user, there is a plain string in the html. There is no way on the user end to access a different translation.
That said, you can get around this by using the locale filter. For instance, you could have a button on your site ‘same page in french’, add a query variable to the link, extract that on the server side, change the language and then generate the same page in a different language (its theme strings, not the actual post content).
Depending on how extensively you want to do this you’re looking at quite some work of quite a lot of work. And you might come to regret that you didn’t use that translation plugin.