I'm developing a very simple web site, with only one web page, but i need the one single page to look as it should in all the browsers. Now, for testing purposes, i have installed a bunch of web browsers on my windows 7 machine (FF, Chrome, Opera, Safari, Netscape etc.) and after doing some markup changes, i've got the same looking webpage on all the browsers.
Now the question is, does "the way" a browser renders a web-page depend on the operating system the browser is running on? Should I install linux (or other os) and test again or it will just be fine?
You should definitely be testing sites cross-platform, others may disagree, but the rendering is definitely different.
In addition to base-rendering, you may also be missing fonts, have alternate fonts with the same name, have anti-aliasing enabled/disabled on different platforms/configurations and much more.
To see exactly what I mean, on a base install of Windows 7 with Firefox 5, OSX with Firefox 5 and Linux with Firefox 5, when using the font 'Arial' or 'Verdana', you will clearly see the differences, even if those fonts do exist on all platforms.
On top of the differences you'd see above, even your positioning could be out if you're using non-absolute values (hell, even if you are using absolute values such as px it could be out!), so it's always best practice to check your sites cross-platform, it's as important as checking cross-browser, in my opinion.