No, you need to do a residency and focus on an area of expertise:
Family Practice- You can provide primary care services for women up to and including childbirth. You can also handle very rudimentary depression and anxiety, but not function as a psychiatrist.
OB/GYN- you can provide a full spectrum of women's health services. You could also theoretically handle rudimentary depression and anxiety, but in the interactions I've had, they tend to punt to psych pretty quickly when they are able to do so.
Psychiatry- you can provide a full spectrum of mental health services, and with the right residency and adequate training, you can become an expert in women's mental health. You will not be providing primary care services as a psychiatrist.
Doing both residencies is complicated due to funding issues. It would also potentially drive you mad. When folks do two residencies (or even two fellowships), it tends to be "holy $hit, I can't believe I did ____" and jump ship. Very few people intentionally do two complete residencies and very, very few maintain and practice in both.
If you are still considering doing it, you can do an abbreviated psychiatry and family practice combined residency in 5 years, if they are still left by the time you reach residency (they've been dwindling).
But you have plenty of time. It is completely level-appropriate to want to do it all. When you trudge through medical school, you start to see what you REALLY like and where to focus. Having too many interests when you start is a much better animal than being two specific ("I'm going to be the best cardiothoracic surgeon in the South East!!!"). The latter folks are the ones who are most likely to get crushed when they go through medical school ("Oh, wow, I don't actually like surgery....").