This sounds like a terrible idea, you shouldn't be wasting schedulers waiting for a http response. It will kill the performance of your system.
That being said, if you have no other options, and you are very careful about what and when you are doing stuff like this it actually is possible using ole automation.
First you need to enable ole automation using
sp_configure 'show advanced options', 1;
GO
RECONFIGURE;
GO
sp_configure 'Ole Automation Procedures', 1;
GO
RECONFIGURE;
GO
Then you can use xmlhttp to fetch the url
Something along the lines of
sp_OACreate 'MSXML2.XMLHttp', @Obj OUT
will create a msxml2.xmlhttp object
You can then call all methods defined in that object such as
sp_OAMethod @Obj, 'open', NULL, 'GET', 'http://myurl', false
You will have to read up on the object and play around with it for a while, but it certainly is possible
xp_fileexistreally isn't meant for this type of operation, its an undocumented proc mainly meant for MS internal usage. As for the 'how', I would really question the 'why', what are you trying to do? This operation sounds like it should be in another layer of the application.