By the closest possible margin, House Democrats’ sales-tax ballot measure has begun moving through the Legislature.

Over the strong objections of minority Republicans, the House Health and Human Services Appropriations Committee voted 8 to 7 in favor of a ballot proposition Tuesday that would ask voters for a temporary three-year boost of three-tenths of a percentage point in the state sales tax. That would up the sales tax rate from 6.5 percent to 6.8 percent.

The plan would raise over $1 billion. About 22 percent would be rebated to about 370,000 households that qualify for the federal Earned Income Tax Credit, and the rest would backfill some of the proposed cuts in the Basic Health Program, the  subsized health insurance plan for the working poor; public health, mental health, hospitals, nursing home and so forth. (UPDATE: Before approving the plan, the committee changed the revenue split between tax relief and health care. The new language says 13.2 percent of the revenue would go to tax relief in the next two fiscal years and rise to 22 percent in FY2012. This means more money for health-care appropriations in the next two years.)

Democrats, including the prime sponsor, Chairman Eric Pettigrew of Seattle, described it as a critically needed source of revenue to patch some of the holes in the human services safety net. Republicans, joined later by Tim Eyman and other tax rebels at an opposition news conference, said people are barely coping with the deep recession and shouldn’t be asked to take on a greater tax burden. If the health care spending is so critical, it should be prioritized higher and included in the budget lawmakers are producing this week, they say.

As you might guess, the tax measure is quite controversial and it’s not clear whether it will pass the full Legislature. Adjournment deadline is Sunday. The Senate is reportedly quite cool on this plan and polls show the public vote would be dicey. GovGreg campaigned last year against tax hikes and has softened her position only a bit.