While I want to stress that leeching on someone else's service like this in a permanent setup is against my personal ethics, I will supply this pointer.
ICMP is often filtered or limited in rate of requests per source address, but HTTP services generally are not. You could use
nping from nmap which (by default) uses TCP handshakes on port 80 rather than ICMP echoes to detect connectivity. As a bonus it works without being root on UNIX systems. The man page describes many, many options for it.
You'll have to either find a static address to a web server with good uptime (I have no idea how to tell if Google changes their front-end addresses) or rely on both a dns server AND a webserver. That might be an issue for your setup.