Hi,
There shouldnt be many reason for this in your situation as the host is directly connected to the ASA interface.
And by default ASA should allow ICMP to its interface if no additional configurations have been made
You can check the output of the command
show run icmp
And see if there are any configurations that deny ICMP to the interface in question.
Naturally make sure that the interface is not missing any configuration and that its enabled with the "no shutdown" command
- Jouni