How do I stop 'Viruses Detected' notifications and remove potential malware from my browser?

If these are Notifications that appear in the upper right of your screen, then go to Safari > Settings > Websites > Notifications, and remove all sites you see there that you have allowed to send notifications to you.

Otherwise if they are appearing on your web pages, then they are just advertisements prompted from the website you are visiting trying to get you to download software that you do no need.

Is this legitimate or a virus in and of itself?

No.

When I first got the notification, I may have clicked on it, because I remember a window opening that gave me the same warning and wanted me to proceed in order to eliminate the virus. I did not proceed. I closed the window.

Closing the window was the correct action to take.

Did I infect my computer by opening that window?

No.

What do I do? How do I delete these ceaseless notifications?

Read Stop unwanted Notifications - Apple Community.

it is very difficult to detect viruses from INSIDE your computer.

No one can detect such stuff form outside your computer.

Unless you deliberately ASKED a service to detect viruses, you may freely assume you are being SCAMMED by lying unscrupulous criminals who will tell you ANYTHING to get access to your attention, then your money, then your computer to be added to a Bot-Net.

NEVER respond to unsolicited reports of anything. If the world is actually coming to an end, you will not be notified by emails and alerts from strangers.

Never provide personal information unless YOU initiated the contact, using fundamental sources you looked up independently, NEVER links, not to "verify your identity" or for Any other reason.

Nothing can become executable on modern MacOS unless you provide your Admin Username and password to "Make it So".

You should be safe. Usually those will take you to a place where they ask for your personal information. Empty your cache and restart your computer