I bought a Korg Pitchblack tuner pedal today. The owners manuel says I should use an adaptor that's 600mA or greater. My Boss adaptor is only 500mA.
I tried the Boss adaptor quickly and it seemed to work fine but should I consider upgrading to an adaptor with more more current?
Can my pedal malfunction or be damaged if I continue to use the 500mA adaptor?

i almost sure you could damage it with MORE current, not with less.
you will do fine with your boss adaptor
^it's the other way around.

Your adapter needs at least 600mA or higher. Running it at a lower rated adapter could damage it.

Also, that seems like an abnormally high amount of milli Amps for a tuner but it that's what it says, then that's what I would use. Just get a One Spot adapter, then you'll be covered if you buy more pedals.