Yes, I would assume that if the US isn't part of the postal union then foreign post services have no way to send letters and parcels to the US anymore unless and until they negotiate service deals with US operators.
The same should apply for sending letters from the US to anywhere else in the world.
The Wikipedia page on the Universal Postal Union states:
"Prior to the establishment of the UPU, each country had to prepare a separate postal treaty with other nations if it wished to carry international mail to or from them."
