Yes, but only in a relatively. Both characters, and most other unpowered superheros, aren’t realistic, because they vastly overestimate how resilient people are. Mark Millar’s Kickass attempts to subvert this.
For the amount of abuse Frank Castle takes on a regular basis, he wouldn’t be able to recover, much less continue hunting down criminals. He’s slightly more realistic in the sense that his day to day activities wouldn’t (usually) be wrecking his body the way Batman’s hopping rooftop to rooftop, and constant hand to hand combat would.
At the same time, recovering from a bullet or even a bad beating will take months, and this is something that both The Punisher and Batman shrug off, as the plot demands.
Anyone attempting to do what Batman does would end up as a subject of a Police taskforce in fairly short order. In contrast, Castle would probably provoke a federal investigation, due to his use of explosives. Neither of these create situations where they’d be able to keep up their crusades for years, much less decades.
So, yes, The Punisher is slightly more realistic than Batman, but not by much. If you want a more realistic comic, then you’re probably going to need to get out of the superhero genre, or look for the comics that specifically address the disparity between someone with minor powers and someone who is, literally, a superhero.
Some quick suggestions on the subject:
Alias by Brian Michael Bendis: this is the original Jessica Jones comic. It’s a good street level view of a character with minor superpowers interacting with normal people.
Powers also by Brian Michael Bendis: This is a series about cops who investigate superpowered crimes and criminals. It’s also very good, and has gotten it’s own adaptation. Still, the comic is worth reading.
If you need more Bendis writing cops, there’s always his run on Sam and Twitch. For reference, it was a Spawn spin off series.
If you’re willing to ditch superheros entirely, then Greg Rucka’s Queen and Country is absolutely worth reading.