Answer by Westley Julio:
The textbook answer is no.
The real world, practical answer is it depends.
Let’s say you have a server and client on the same broadcast domain. The client software requires that you input the server address in order to enable a client-application to function. While the client device can technically communicate directly with the server at L2 only knowing its MAC, the ‘data’ won’t be generated unless you satisfy some requirements at L7 (if OSI is your frame of reference).
Think about the basic ping command. For example, “Ping 192.168.0.2″ – Your computer will use ARP to automatically resolve the IP into a MAC. There isn’t a provision to ping a MAC (that I’ve come across anyway). So, the ‘data’ in this instance requires an IP, even though the network does not.
You could use a packet tool to directly generate an ICMP packet (for example) and send it off to the network, or replay something previously captured, which would bypass the IP address and be strictly L2. However, nobody does this in practice.