The new Google APIs have a common API infrastructure where client libraries are always updated. There is such a client library for Dart (http://docs.google-api-dart-client.googlecode.com/ git / index.html).
However, it appears that Google Spreadsheets is still using an older API framework called GData. There is no GData client library for Dart that I know of.
Therefore, I think the most sensible thing is to make raw HTTP calls. I'm not sure if you are trying to do this on a client or server. If you are trying to do this on the client, you will need to make sure that Google spreadsheets support CORS.
If you use a different programming language on the server, it looks like the GData client library for that language. See (https://developers.google.com/google-apps/spreadsheets/#setting_up_your_client_library).
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