The resulting inflation would not be nearly so great as to cancel out the gains made by those minimum wage workers.

One must take into account both that the proportion of the cost of goods that is spent on labor is quite small and that the increased income for these individuals almost entirely translates into increased demand for goods.

Let's look at 2 very different goods- landscaping and big box stores.

This might sizably increase the cost of landscaping because labor is a large proportion of a landscaping company's expenditures but, since few minimum wage workers can afford to hire landscapers, America's minimum wage workers would still benefit from their wage increases. The real risk is that the landscaping industry could take a big hit because fewer people will pay the increased cost and this could result in unemployment for some of the workers.

A big box store is a wholly different animal- it primarily serves low income workers thanks to the low prices enabled by economy of scale benefits that return to purchasing lots of goods at once. Labor is a very small portion of the costs, so the price of goods would go up very little. Since the consumer base it serves would see a sizable increase in expendable income, the demand for the goods would increase and the store may benefit from hiring more workers to meet this demand.

I think OP assumes the big box store in this example would simply increase prices by the proportion that wages are increased, but this would not be a good idea in a competitive marketplace. The risk would be that perhaps a world in which Wal-Mart no longer has to offer low prices could come into fruition and in such a world we would be fucked. And perhaps concentration of capital would occur due to a minimum wage increase- people often claim that this puts small businesses out of business- so this is worth thinking about. However, even with the disturbing concentration of capital already, it is not as though Wal-Mart could remain competitive if they just increased their prices by 50% in response to having to pay workers $15 instead of $10 an hour.