Try this:

• download the Firefox installer and save the file to the desktop
  *https://www.mozilla.org/en-US/firefox/all/
• start the installation with a double-click on the desktop icon

It is possible that security software (anti-virus, firewall) is causing the problem. Try to disable security software temporarily to see if that makes a difference.

I followed your link, scrolled down to "English" and chose "Windows 64-bit". When I clicked on it I got "Secure Connection Failed". The site is Mozilla.net - how can it not be secure?

There actually are many possible reasons Firefox might have trouble making a secure connection to the site, it's not always because of a problem with the site itself or its certificate.

There are two main error pages:

• "Your connection is not secure" -- page generally has an "Advanced" button to get more technical information.
  • What do the security warning codes mean?
  • How to troubleshoot security error codes on secure websites

• "Secure Connection Failed" -- typically has scantier information about the problem and if the padlock appears normally, can be taken to mean "connection failed"
  • Secure connection failed and Firefox did not connect

Mozilla has installers on other servers, as well. For example, if you want a Windows 64-bit installer for U.S. English on a server that supports HTTPS:

https://archive.mozilla.org/pub/firefox/releases/53.0.3/win64/en-US/

(For other languages, click "up" using the .. link.)

I clicked on the link you provided, then clicked the link that appeared there. Got "Secure Connection Failed".

Then I pasted your link into Edge, clicked the link provided and got "Hmm...Can't reach this page". Clicking on "details" got this: "Error Code: INET_E_DOWNLOAD_FAILURE".

How about without the "s"

http://archive.mozilla.org/pub/firefox/releases/53.0.3/win64/en-US/

(Obviously you shouldn't get a secure connection error with a HTTP download link.)

Sometimes inability to download EXE files indicates a malware problem. Unfortunately, Mozilla doesn't provide other formats, such as .msi installers. If you haven't checked your system for malware recently, please see this article for suggested tools: Troubleshoot Firefox issues caused by malware.

A logical answer, thanks. I did a full scan with Zone Alarm Security Suite and with Malwarebytes yesterday. Neither found any problems.

How about without the "s" did work! At least it got me far enough into the process to learn the real problem...

Windows Defender considered it dangerous and warned against installing. When I insisted, it said the program can't run on my computer ?!?

As soon as I figure out how to disable Windows Defender I'll try again.

Well, as far as I can tell I already had disabled the firewall and virus protection because I have my own. Could it be right? is it possible the latest version of Firefox can't run on a Windows 10 machine?

PaulAnthony70 said

How about without the "s" did work! At least it got me far enough into the process to learn the real problem...
Windows Defender considered it dangerous and warned against installing. When I insisted, it said the program can't run on my computer ?!?

That's truly strange. I don't know why that would be.

You can upload the exe file to this site for a double-check:

https://www.virustotal.com/

Could it be right? is it possible the latest version of Firefox can't run on a Windows 10 machine?

No, I think there would be hundreds of posts about it if it were a general problem.