Scotland is part of an existing member state of the EU, but it is not a member state of the EU. Were it to become independent, it would likely have to join the EU as a new member state.
There's perhaps an argument to be made that if the UK broke up, each successor would have a claim to be a successor to the UKs treaty obligations and benefits in general, which might carry some weight (though would be extremely complicated for a treaty arrangement like the EU) if it happened before the UK formally started the process of withdrawing from the EU (if it happened after that, even viewing Scotland as a successor to the breaking-up-UK would leave it a short-term member of the EU in the process of exiting under the Article 50 process, inheriting the exit date that the UK has, and needing to negotiate its own exit agreement if it wanted anything but the default exit terms.)
Now the next referendum will be a lot more significant since it'll effectively be a UK vs EU decision. Either you follow the English into isolation or you suddenly have a very real border between England and Scotland with customs&immigrations checkpoints and similar things.