In order to do something meaningful, applications will need to interface with the OS. Since system calls and user-space infrastructure look fundamentally different on Windows and Unix/Linux, having different formats for executable programs is the smallest trouble. It’s the program logic that would need to be changed.
(You might argue that this is meaningless if you have a program that solely depends on standardized components, for example the C runtime library. This is theoretically true – but irrelevant for most applications since they are forced to use OS-dependent stuff).
The other differences between Windows PE (EXE,DLL,..) files and Linux ELF binaries are related to the different image loaders and some design characteristics of both OSs. For example on Linux a separate program is used to resolve external library imports while this functionality is built-in on Windows. Another example: Linux shared libraries function differently than DLLs on Windows. Not to mention that both formats are optimized to enable the respective OS kernels to load programs as quick as possible.
Emulators like Wine try to fill the gap (and actually prove that the biggest problem is not the binary format but rather the OS interface!).