There is no simple way to get something like the Net Map view using the API, you would need to parse through the networks and network containers in the response to a GET request.

If you want to know if a new network block will fit in a container, however, you can use the next_available_network function. If you run that function on a Network Container and set the CIDR field to the size of the network you need, you will either get a network block returned successfully or you will receive an error if there isn't enough space.