Welcome to the forum!
I wonder if your project is drawn inside the page border? If it's not, it might not print.
You probably noticed the page border the first time you opened Inkscape. Most people seem to think it's a rectangle which they don't want. It can be switched off, so if you had done that, you might not realize the drawing is not inside the border.
That's only a guess, however.
Usually when you click the Print button, or File menu > Print, a print dialog should come up, where you need to choose which printer. For example, I don't have any printers attached to my computer, however, the Print dialog offers me 2 choices. Since you mentioned a slightly different series of actions, i.e. expecting it to start printing when you "tap" the Print button. And only later checking on which printer what chosen, it makes me wonder whether you saw that dialog.
Or do you mean you clicked (or pressed) a print button somewhere else, besides Inkscape?
