I think the closest you can reliably get is to determine what version of the CLR is required. You can do this by using ILDASM and looking at the "MANIFEST" node or Reflector and looking at the dissasembly view of the "Application.exe" node as IL. In both cases there is a comment that indicates the CLR version. In ILDASM, the comment is "// Metadata version" and in Reflector the comment is "Target Runtime Version".
Here are examples for a .NET WinForms application named WindowsFormsApplication1.exe:
By finding the reference with the highest version metadata you can determine what version of the Framework that reference came from, which would indicate that you need the same version of the Framework installed for the application to run. That being said, I wouldn't treat this as 100% reliable, but I don't think it will change any time soon.
From code you can use Assembly.ImageRuntimeVersion but by looking at the file probably the best thing to do would be to use reflector and see which version of mscorlib is being referenced.
Edit: Even better would be to use ildasm, open your assembly and then view the manifest for the assembly. The first line of the manifest will tell you the exact version of CLR that the assembly was built for.
You can use a tool called CorFlags.exe. It has been around since .NET 2.0, and I know for sure that it is included in the Windows SDK 7.0. By default (on Windows XP Pro) it is installed to C:\Program Files\Microsoft SDKs\Windows\v7.0A\bin\CorFlags.exe. Provide it with the file path to a managed module (without any other command-line flags) to display its header information, which includes the version.
Keep in mind that this utility is designed to modify the PE32 header of a module, so don't use any of the flags until you read the documentation carefully.
Or you can just find out which reference of System.Core it has. That will tell you the .NET Framework version this app is using. For 2.0 the version of System.Core will be 2.0.xxx.xxx. For 3.5 the version will be 3.5.xxx.xxx, etc.
You can now use ILSpy to examine the target framework of an assembly. After loading the assembly, click on the root of the assembly node, and you can find the information under the TargetFramework declaration:
Using Notepad, three decades old, 200kb in size, preinstalled tool:
open application with notepad appname.exe,
search for word framework,
repeat last search with F3 until .NET Framework,version=vX.Y shows up
if nothing found (versions below 3.0) search for v2. ... still 100 times easier then installing gigabytes of dot net analyzer tools and garbage studios.
update: Thought and Marcus suggested in comments that search term could also be netstandard or netframework
For newer .NET 6.0 applications the notepad method did not work for me, but if there is a *.deps.json file next to the exes that's a big clue - often contains the version in form of the following section
Edit: following up, you can also look for a pair of .exe and .dll files with the same name, try opening the DLL file in notepad and search for NETCoreApp, you should find something like this: