Supporting the NATO mission politically is different from supporting the mission militarily. Same happend with ALLIED FORCE in 1999 (Kosovo/Serbia) - some NATO members opposed it, but did not 'break silence' but would not commit forces either (eg Greece).
There is no voting system within the North Atlantic Council, just concensus. Inasmuch as it is fun to have a pop at NATO, getting 28 Countries to agree politcally to this mission is frankly astounding. Almost as astounding as getting the UNSCR to agree such a robust resolution (UNSCR 1973). Of course Russia and Chuina abstained. Ongoing politcal turmoil in North Africa is a great for the gas business - the Russian gas business.