Except the minimum wage hasn't actually increased anywhere but Seattle, Washington(and even there it's still being phased in), and more-over, one of the big principles that undercuts this argument is: "once you can automate away a job, is there any wage at which you wouldn't?"
No, there isn't any wage at which you wouldn't - and it's been happening right under our noses for thirty-forty odd years now. Most people don't notice it because "automation takes away jobs" is virtually always assumed to mean "low education, low or no skill, rote and/or repetitive" jobs.
But the microprocessor revolution changed all that. The skilled master machinist has been replaced by an unskilled worker who loads and unloads a CAM machine. The draftsmen that, under the direction of an engineer, created and maintained the drawings the machinist worked from has been replaced by a CAD program used directly by the engineer. The engineer himself has been partly replaced by electrons too... instead of spending weeks with slipstick working out a stress calculation, now sets it up in a day or two on the appropriate software, clicks the mouse, and it's finished before he gets back from freshening his coffee.
And that's just one example, consider the business my wife works at... Thirty years ago, and at a tenth the size they had a full time accountant and two full time bookkeepers (plus data entry clerks and file clerks) - now they have an (almost) full time accountant, the bookkeepers (along with the data entry clerks and the filing clerks) having been replaced by a POS system.
When it's skilled, or especially when it's white collar, we call it "productivity improvement"... but we should call a spade a spade. It's automation.