I'm interested is there a possibility that could allow to display website's content or to say exactly an HTML through a specific browser installed on the web server?
I mean something like a module for a web server may be, that can display the website's content through the built-in browser, ignoring the clients browser?
If this possibility really exists, so I don't need to adopt my HTML to different browsers.
No, there isn't any web server module that takes control over the client's computer.
Depending on your situation I suppose you could replace the HTML page with Flash, Silverlight or a Java applet if that would make things easier.
Well, you could make some pdfs or images if you really want a specific rendering, but it's really not a website anymore at that point. You could also beg your users to use a particular browser, but they'll probably ignore you. The world spends (wastes) gazillions of dollars on good, standards-compliant HTML and CSS for a reason.