A search for muon neutrinos from neutralino annihilations in the Sun has been performed with the IceCube 22-string neutrino detector using data collected in 104.3 days of live time in 2007. No excess over the expected atmospheric background has been observed. Upper limits have been obtained on the annihilation rate of captured neutralinos in the Sun and converted to limits on the weakly interacting massive particle (WIMP) proton cross sections for WIMP masses in the range 250-5000 GeV. These results are the most stringent limits to date on neutralino annihilation in the Sun.