It depends on your license. If you have an advanced license you should have a tool called Erase. Use this tool with both of your datasets and you will get the correct result.
If you do not have that tool, please let me know, and I will edit my answer to explain a workaround!
For those who do not have access to the Erase tool with their license, here is a workaround:
Step 1: Use Union with both of your files as the input
Step 2: Do Select by Location, selecting all the features from your original dataset (the green one in your screenshot) that are within the bigger dataset (the output from union, that is). Now, this will select all of your features in the union dataset that have been originally there, and it leaves out the parts that have been newly added through union.*
Step 3: Start an Edit Session and delete all of the selected features. This will leave you with the "white parts" that you were looking for.
*I said within but the terminology in the selection tool might differ slightly. I cannot check right now as my license has expired. So you might have to try some of the many keywords in the unnecessarily confusing tool! Should you do this in ArcPy it would definitely be "within" though!