That depends on your purpose. From a professional point of view as an IT person, know Java is very important. I assume, though that's not a consideration for you, given that you haven't yet.
So, I'll rephrase your question as "Is learning Scala enough, or do I need to learn Java too to program in Scala?".
Learning Scala is enough, and I think Programming in Scala does a great job of doing it, but, then again, I know a lot about Java.
There are two things about Java that you might miss, though:
1. The whole Java infrastructure.
You may want to make "jar" files (or even "war" files or similar stuff), run it on web servers or computers without Scala installed, and things like that. This kind of stuff is not covered by Programming in Scala. Then again, this kind of stuff is usually learned through experience, not books.
2. Java libraries.
There are many, many Java libraries that you might be interested in. There is no shame in that. After all, Scala itself uses Java's standard library extensively, and there are tons of good stuff out there. There is no sense in spending days to display a simple graph of data when there are libraries which will display your data in dozens of different graphic formats.
It's important that you are able to read API documents, and understand simple examples.
I don't think either of these two things require a more formal Java education. If you can get by, it should be enough.
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Daniel C. Sobral
Something I learned in academia: there are three kinds of academic reviews: review by name, review by reference and review by value.