If a destination address is not in the ARP table for IPv4 on a data-link protocol that uses MAC addresses, then, yes, it would require an ARP request to resolve the MAC address from the IP address. IP (ICMP is an integral part of IP) has no knowledge of the underlying data-link protocol, so it knows nothing of MAC addresses or ARP.
Remember that the network stack is layered, and there are many different data-link protocols, and IP does not know or care which carries it, nor does the data-link protocol know or care which network protocol is in its payload. Even with something like ethernet that needs ARP for IPv4, ARP is not used for other network protocols. For example, IPv6 on ethernet will use NDP because ARP uses broadcast, but IPv6 does not have broadcast.
Once an IPv4-to-MAC address resoultion is in the ARP cache, it will stay there for a specified time, so an ARP request may not be needed. When a host sends an ICMP echo request, and it needs to send an ARP request, the recipient of that ARP request can use the ARP request to update its ARP table, so it may not need to send an ARP request to be able to reply; it can simply look in its ARP cache.
