If the minimum wage rises marginally above inflation then of course it won't result in unemployment as rising prices, offset rising wages and therefore it's hard to argue against it. The problem come when you have a hike in the minimum wage, like the proposal to raise the minimum wage to 15 dollars an hour. However there can still be some problems say the economy is in recession, and there's a fall in demand. You have to options, reduce wages or cut jobs. In the UK wages were quite flexible during the great recession and so offset a fall in jobs. In the US unemployment rose to a higher level than in the UK. Reducing wages is probably better than cutting jobs, but if your workers are on a minimum wage then there's no flexibility.
My take is that it would ignore the laws of economics to say that any change in the MW would have no effect on MW employment.
On the other hand, economics also recognizes the concept of demand elasticity -- the fact that for some goods a small change in price has a big effect on demand, while on others, a large change in price only has a small effect on demand.
The demand for low price labor seems to be fairly inelastic -- that is, an increase in the price of labor (e.g. the MW) has a moderately smaller effect on demand for low end labor.
Thus, for example, we have seen periods (e.g. the mid 1990s) of significant MW increases with little or no apparent effect on MW employment. However, there is certainly a level where the higher cost will have a greater effect on employment.