Yes.
The only compatibility difference is when you go backwards, from 2007 to 2003.
07 by default saves in .docx format, while 03 saves in default by .doc format. However, 07 DOES support .doc just fine.
To get 03 to support .docx (just for future reference even though you're in the clear in your current situation) there is a "FileFormatConverter" you can get on the Microsoft web site, which will auto-convert .docx files to open in 2003 without any issue.
But in short, you are okay. Like I said, going from 07 TO 03 is when you run into a compatibility issue.