then abortions are available to all without any restrictions.
In most countries in Europe, they are. Or token restrictions.
which crime is reported falsely most often
It's rare we agree. The women's movement say that false allegations are (in their frequent words) "vanishingly rare". The so-called 'men's movement' (losers the lot of them) say it's as much as 30%. Neither are right and neither of those stats have any basis. The UK government say that 9% of (all) prisoners are probably innocent of the crime for which they're convicted, the Henriques Report (into the false allegations of a turd called Carl Beech) suggested 7%. I wouldn't like to guess at a figure however yes, false allegations happen.
That's veering away from the topic, since if abortion is allowed in some places after rape, who decides? If the doctors wait for a conviction, the baby will already be born. If they do it after an arrest and the person isn't charged, are they guilty of murder?
Let a woman decide if she needs an abortion.