@JTT,
Not unless your computer is already compromised, and in that case it would be easier for the malware to just phone home. That's the long answer, the short answer is really: no.
It's just a text file and doesn't hold much information (each request to the server that originated the cookie sends the cookie, so by necessity it needs to be very small). Usually it will just contain a unique numeric id that the site uses to reference data stored at the site.
For example, a2k uses a cookie that just contains your session id, which is just a long random string of numbers and letters that is tied to a session on our end.
The way the internet is designed is "stateless", which means the core protocols weren't meant to store "state". Each request was anonymous and blind to previous requests.
Storing "state" means stuff like knowing that you are logged in (instead of requesting a login on every page load) and cookies are just a way to store application state.
So as an example, when you log into a2k it sets a cookie with your session id, and with each request the server looks for that session id and if it exists and matches what you sent it keeps you logged in. There is another cookie (autoLogin cookie) that stores your login to automatically log you in if your session dies.
Browsers are configured to only let the domain that set the cookie read the cookie, so each site can only read the cookies on its own domain by default. It's theoretically possible to put malware on someone's computer that circumvents this kind of restriction but if you can do that you already have full control of the computer and you don't need to use cookies to get the information out.
But to generalize your question a website can do limited things to tell whether you have visited a particular site. For example, if you have a site in your history the "visited" links are often a different color. A site can link somewhere and then use javascript to check what color the link is and this way detect whether you've been to that particular url. It can't search your history and has to load a list of links to check. Here is a proof of concept:
http://www.debugtheweb.com/test/cssvisited.htm