I started medical school while our kid was in preschool, but I'm the mom. Like it or not, whether or not you are the one who is pregnant does make a difference 😉 Many of my classmates became fathers before/during medical school, and as far as I know, in all cases they are happy parents.
The real hardship is financial, especially if your spouse is currently working, or if she has loans to pay off and not working will cause her to rack up a ton of interest. Some spouses have made that work by working nights/weekends so they can be home while the student is in class. In my case, my mother moved in with us, so my DH has kept his job, but non-nuclear families aren't for everyone.
I personally don't think becoming a dad in medical school (or just prior) is a big deal, as long as your spouse is supportive, the two of you have realistic expectations, and are well educated about what lies ahead and plan well, realizing that those plans will likely not work, so you also must be flexible. Read blogs (I read Panda's blog back in the day), and describe to her how much work/time/$$ is involved.
Your school should be supportive and help you deal with, say, delaying an exam because you went to watch the birth of your child. On the flip side, you and your family need to deal with the fact that you may miss birthdays and other special occasions because you are on call, because you need to study, etc. That will carry into your medical practice, so this is not a "when" but "if" issue.
This is great for an independent spouse who makes new friends easily and does her own thing. This is terrible for a spouse who is accustomed to discussing every tiny decision with you, who expects to spend several hours each evening with you, who expects you to drop everything each time the baby kicks, attend every prenatal appt, hold her hair when she pukes. This is also terrible if YOU want to be the guy who does all these things, as well as being present the first time the baby smiles, etc. I had to let go of a lot of control in parenting, because I was no longer the primary caregiver once I started med school (I became one of 3 caregivers). She'll need to be the main decision maker regarding when the baby sleeps, what the baby eats, discipline, etc, because she has to deal the consequences of those decisions more than you do.