It is technically "unlawful" for a US citizen to enter or leave the US without "bearing" a US passport (or alternative document as allowed by the US government, mostly border crossing cards for the land border), under to 8 USC 1185(b). However, no law or regulation provides for any penalty for violating this law, and violating it will not impact your future ability to get a US passport or obtain any other services.
Since the US does not have physical exit controls, nothing will stop you from traveling out of the US without a US passport, as long as you have the documents to enter the destination country (e.g. with your non-US passport). If you are traveling by air, the airline will almost certainly only care about your ability to enter the destination country. It is theoretically possible for CBP to make random checks of departing passengers, but this is extremely rare and I have never seen it.
Of course, once you are abroad, the next question is about how you would return to the US. Assuming you have to fly back to the US, the airline will not let you board without a US passport, since you don't have a green card anymore and presumably you don't have a US visa or ESTA or any other document accepted for travel to the US. You could apply for a US passport at any US consulate abroad, but that may take some time. (Although, given your situation, I am assuming that leaving the US to visit your father is very urgent, but coming back to the US is less urgent, so I am guessing that, in the worst case, having to be stuck abroad for a while is still worth it.) Or, if your passport is received in the US, a friend in the US could try to mail it to you abroad.
Another option for returning to the US would be, if you can get a visa or visa-free access to Canada or Mexico on your non-US passport, you can travel to Canada or Mexico and then enter the US by land, with your Certificate of Naturalization and other IDs to prove your US citizenship. This would again violate the law mentioned above, but they cannot deny entry to a US citizen. So if they are satisfied with your US citizenship (which probably involves temporarily detaining you while they verify your citizenship, and giving you a lecture), they have to let you in, and there are no other penalties or other consequences for entering without a US passport. (They might impose a fee for such cases in the future, but as of now they haven't done so.)