A commenter on another site pointed to this Hudson GRL 2011, which attempted to answer this very question with better science by including changes in cloud cover and the seasonal variation of solar radiation.
He concludes that the increase in forcing due to Arctic sea ice over the period 1979-2007 is about 0.1 W/m2. (My estimate was 0.12 W/m2, which was closer than it deserved to be.)
By comparison, manmade CO2 forcing is now about 1.8 W/m2, and the forcing from all manmade GHGs is about 2.8 W/m2.
(Incidentally, that means, if you lump all GHGs into a CO2-only picture, the effective atmospheric CO2 level is about 470 ppm, as of 2010.)
"...a complete removal of Arctic sea ice results in a forcing of about 0.7 W m−2, while a more realistic ice-free-summer scenario (no ice for one month, decreased ice at all other times of the year) results in a forcing of about 0.3 W m−2, similar to present-day anthropogenic forcing caused by halocarbons. The potential for changes in cloud cover as a result of the changes in sea ice makes the evaluation of the actual forcing that may be realized quite uncertain, since such changes could overwhelm the forcing caused by the sea-ice loss itself, if the cloudiness increases in the summertime.Marc Morano and Steve Goddard are desperately trying to distract attention by highlighting the gain in Antarctic sea ice. The problem, of course, is that the Arctic is melting much faster than the Antarctic is gaining ice: -54 Kkm2/yr compared to +14 Kkm2/yr, since the beginning of satellite records.
Added: Then there is the fact that Arctic sea ice tends to be thicker than Antarctic sea ice: about 2-3 meters versus 1-2 meters, says the NSIDC. So a loss of a square meter of Arctic sea ice is a larger volume loss than is the volume gain of an Antarctic square meter of sea ice.