Example bash on argument 1. It checks if HTTPS is up and if the domain-name is on the site. Most sites have the domain name on the front page somewhere: /usr/local/nagios/libexec/check_http -H $1 --ssl --onredirect=follow --string "$1"
If log file you can run grep on the following keywords: grep -i -e "critical" -e "invalid" -e "warning" [LOGFILE.log]
Keywords: check_http