The request had been expected after a mid-level appeals court ruled last week against the measure. It would put a 16-ounce size limit on non-diet sodas and other high-calorie soft drinks sold at many eateries.
A judge had blocked the size cap in March, shortly before it was set to take effect at restaurants, movie theatres, food carts and some other establishments.
The city sees the regulation as a needed, if novel, step in fighting obesity and related diseases.
"This is a pressing public health matter that deserves careful review by the state's highest court. The Board of Health must be able to combat the growing obesity epidemic, and there is clear precedent for the board to have that authority," the city's chief lawyer, Michael Cardozo, said in a statement Monday.
The American Beverage Association has helped lead the fight against the regulation. The soft drink manufacturers' group says it's confident last week's ruling will stand.
"We look forward to a final resolution of this issue," the organization said in a statement Monday.
The beverage association and other critics have argued that the health board overstepped its bounds, that the regulation would crimp consumers' choices and that it's unfair because it exempts supermarkets, convenience stores, alcoholic beverages and milk-based drinks.
There's no set date for the Court of Appeals to decide whether it will take the case.