Depression Supporters Support Group
This community is dedicated to spouses, children, relatives, co-workers and others who are actively supporting someone suffering from depression or other serious mental illness. Depression can bring about intense sadness, melancholia or despair that has advanced to the point of being disruptive, which can have a grave impact on the ability of others trying to help.
I'm not in your exact situation, but I do have a lot of experience dealing with depression (both my own as a teenager and now my spouse's).
If you're looking for a way to connect with your daughter that doesn't sound like you're checking up on her or judging her by asking how things are going, try asking about something more specific. If she's in college, ask how one of her classes is going. If she's working, ask how things are going at work. If she regularly spends time with a specific friend or boyfriend, ask how that's going.
You can tell a lot about how things are in general by those other topics. It also takes the focus off of her and makes it more about the other things going on in her life. Sometimes calling to talk about something you're struggling with can help get her talking about her issues - it creates an open safe environment. You share something and she shares something.
You could also call her up and make up an excuse to go do something together and casually chat about things while hanging out. Let her know you're there for her and you're thinking about her without making her feel like she's under a performance review.
My spouse told me once that asking him how he's doing made him feel guilty because his answer would inevitably be negative and would remind him of his "failing" to be doing better. Of course he wasn't failing at anything and had nothing to feel guilty about, but that's how depression works. It makes some people feel worthless and then guilty for not being more worthwhile.
It's a terrible and heartbreaking thing to be dealing with. My support goes out to you. The two of you will get through this.