After looking into the Ping - ICMP port number, I have come to the conclusion that there are two theories on which port Ping uses, and no conclusive evidence that either is 100% correct:
One group of people believe Ping does not use an IP port. Here's the argument that backs them up:
"Ping does not use a port number in the way we normally think about it. A port is associated with a socket that lives on the transport layer. You can't ping a specific port number as it is a completely different protocol/transport type from TCP, UDP, GRE, etc."
Then there are those that think Ping uses IP port 1.
Their reasoning is something like this:
"I'm not saying that ICMP uses either TCP or UDP, what I am saying is it sits on top of IP and is assigned IP protocol number 1. As another example EIGRP doesn't use TCP or UDP, and is assigned IP protocol number 88."
Ultimately, you're going to have to chose between which of these two camps makes most sense to you, as documentation on the matter is pretty scarce!