What, if anything, can I do if told that I may not be able to go home from a stateside deployment for the birth of my first child?

I missed 2 of 3 births due to deployments/mission 1st.

Unit Commanders hold the approval authority for leave and passes. You can use your Commander’s Open Door Policy to discuss it, but beyond that you have to remember that your Commander is balancing mission accomplishment and troop welfare at a level you’re not seeing as an E-4. Commanders don’t callously sweep aside requests involving child birth, as it is a significant event.

Yeah, pretty common. Soldiers missing births, holidays, anniversaries, etc... are so common it's basically a cliche at this point.