No - IIS won't know if a site is "down" unless IIS is turned off (in which case, how will it tell you), or you've manually turned just that site off, in which case, you know you did it. Now, if the site is unreachable because the internet connection is down, how would IIS know about that? If the site is so slow as to be unusable, because the server is getting DOS-ed, how could it tell you? If the site is unreachable because the server itself went down, how could it tell you that?
That's all just a long way of saying you have to do monitoring of services from a different machine than the one providing the service. Those 3rd-party monitoring sites are great for that. Some are free, some can get kind of expensive depending on your needs.
You could also script something up yourself, but you'd have to spend time writing it, testing it, debugging it, and then host it on a different server at a different location on the Internet, and hope that it doesn't go down.
Unless you're a big operation, with a reason to invest a lot of money into a monitoring infrastructure, just use a free or cheap third-party.