I was in the process of applying for residency (partner visa) in australia last September when I had to cancel my application to come back to the u.s. in order to have my baby. I am interested in moving back to australia now, but I was told as I left the Australian airport that I could not return for three years. Is there any way around this rule?

Who told you this

The person at the gate in melbourne airport ; I showed them my passport and she pulled me over to the side and told me "do you understand that you will not be able to come back into the country for three years?" and I said "ok" and went through to get on my flight to auckland first and then on to L.A.

Like the airline's ticket agent or someone from Australian Immigration came all the way down to your gate to tell you this.

Don't believe anything a ticket agent tells you, they know nothing of your passport situation and don't know all the laws about immigration. From what I read you can cancel a visa application without consequences, except losing the application fees. From your description above you never received a spousal visa and simply cancelled the application.

Contact the Australia consulate in the US and ask about re-applying, I think you will be allowed to.