tabs and internet?

Someone may help you if you explain your problem more clearly!

Which browser (Safari, Firefox, Chrome, etc.)?

What does "quit my internet" mean?

Well now we know it's safari.

Right click is for a contextual menu but what "little icon" are you pointing at? And I still not sure what "light underneath the icon" means. We have a terminology problem here I think.

I opened safari to try to understand what you're talking about (my browser of choice is firefox). There's the Quit menu item. There's a quit menu item in the dock's contextual menu for the safari icon in the dock. And there is that little light blue triangle under the dock's safari icon indicating safari is currently launched. Ah, ha, is that the "light" you are referring too?

If it is then selecting that dock's contextual quit item should cuase safari to terminate and be deleted from the dock unless you are keeping it in the dock, in which case your little "light" should disappear. I think we're on the same wavelength now 😉 (I hope)

So if we're talking about the same thing then you are saying selecting that dock's quit for safari doesn't make safari quit. Under those conditions does it quit when you select the quit menu item in the main safari menu in the menu bar at the top of the screen (i.e., command-Q)? I would think it wouldn't if the one from the dock doesn't work.

Before we go further please confirm my questions above (is it the dock quit and does command-Q work).

So my assumption is correct, you can't quit safari. It doesn't matter whether it's from the dock or menu. I also presume then the quit menu item in safari is grayed out? Is it?

Do you have any extensions installed? If so try running without them (if not skip to the next expirement). I would suggest using the safari preference's extension tab and turning them off. But there may be a chicken-and-egg problem to get the preference to stick if you need to force quit safari because you can't quit it normally (which is the problem you are trying to solve).

An alternative approach would be to go into your ~/Library/Safari (~ is your home directory) and just drag the entire Extensions folder in there on to your desktop and then launch safari. It wouldn't (shouldn't) have any extensions now. Does that allow you quit safari normally now?

If no, next experiment. Drag your ~/Library/Preferences/com.apple.Safari.plist file on to your desktop (so at this point you should have the Extensions folder and plist on your desktop) and launch safari. Work now?

If still no, drag the Extensions folder back into the ~/Library/Safari folder (so now only the plist remains on your desktop at the moment). Now drag the entire ~/Library/Safari folder to your desktop. So now at this point the entire Safari folder from ~/Library and the com.apple.Safari.plist are on your desktop.

You are now in a state as if safari has never been run (ignoring caches which I believe we can). A launch now should be equivalent to running safari the first time -- no preferences, no extensions, no personal bookmarks. Hopefully you should be able to quit it now. But let's see what the next steps are after these experiments.

Reminder - Do loose you original plist and Safari folder which should currently be on your desktop.