From what I can find it looks like that's simply not an action supported out-of-the-box.
The closest thing is
Environment.GetEnvironmentVariables(EnvironmentVariableTarget.Process);
Unfortunately that seems to be "locked" to the current process. If there were just a way to set what process to target you'd be golden.
Fortunately, Microsoft provides a "reference source" you can use to see what the underlying code in .NET is (in this case http://referencesource.microsoft.com/#mscorlib/system/environment.cs,20e3d8aa4eb8f4b1). If the function supports doing what you want you could make your own wrapper for it.
If you trace that down you find a Win32 call in an unsafe context is ultimately responsible: http://referencesource.microsoft.com/mscorlib/R/c48702b7df790f4c.html
It looks like even that call is locked to the current process, so you'll need to write your own code for it. Here's an example in C (or C++, can't tell lol):
http://www.codeproject.com/Articles/25647/Read-Environment-Strings-of-Remote-Process
You can translate the gist of that into unsafe C# and do the same thing using DllImports, etc. as necessary.. Sorry the answer isn't that easy, but it sounds like a fun project to me!
Edit: Oh I love how the linked possible duplicate did the same research that I did, found the same things, heh.. Just know that I didn't just copy-paste, did the research myself..