my partner has very sever depression and I knew this from the beginning of our relationship. When we first got together he was actually going through a really bad spell, so I can understand your worries.
My advice would be to not push him to talk about it too much or to make it into a big thing right now if that makes him uncomfortable, but at the same time to acknowledge that it is a factor in his life and now therefore in your relationship. If you don't say anything at all then it can become a huge elephant in the room, so to speak, for both of you.
Some people have a lot of shame and guilt associated with their depression, and find it super uncomfortable to talk about or even admit to. But he's already opened up to you to tell you that he's going through this, so when you next see him, acknowledge that and thank him for trusting you enough to tell you. Let him know that whenever he's ready to talk to you about it, you're willing to listen and that you're not ashamed/freaked out by him.
Then if he does want to talk, just listen: don't necessarily try to offer 'solutions' or suggest ways he should fix himself, and if it really is something you've not experienced, don't offer 'oh that sounds like when I feel x y z' kind of responses. (I've learnt this the hard way, btw! It can be hard to repress your instinct to want to relate someone else's experience to your own, as a way of trying to convey empathy. I'm someone who always wants to 'help' by offering solutions... I have realized that this is absolutely not helpful!) The best thing to do is just let him talk when he needs to, and be willing to listen without judgement.
In the long term I've found that the more open you both can be about it, the easier it is to deal with within a relationship. With my husband, there have been times when he's been fine and times when he's been really bad. We check in with each other I take the time to ask him how he's doing, and give him a space to talk if he wants to when he's going through a bad spell. We have also briefly met with a counselor together as a couple actually during a period where he was not doing too badly and we were feeling very close anyway -- just to get another perspective on how we deal with it together as a couple. The counselor has some good advice and we both found it useful, although I know not everyone is comfortable with that kind of thing.
But I would say also, make sure to look after yourself. By which I don't mean avoid having a relationship with someone who has any history of depression or current depression! That seems very extreme to me... (and seeing so many people try to hide their depression or aren't diagnosed, if you tried to avoid dating anyone with any kind of depression you'd probably be single a long time!) But rather, if your partner is going through a bad spell and you find it difficult to deal with, make sure to take time for yourself, recharge your batteries, go off and do something for yourself for a while.
And above all remember that if they are sad or feeling down, it isn't a reflection on you or the relationship you share. It's not your fault if they are sad, any more than it would be your fault if they had a broken leg or diabetes or any other kind of illness.