Em, no - that can only be done in internal store "data" directories which are usually formatted with UNIX filesystems and are by default secure (and cannot be accessed by other apps at all).
WhatsApp is storing to external (on most devices FAT) storage (which was SD card on older devices, it's usually a separate directory/partition on newer ones) which does not have any ACL-like system due to FAT backwards compatibility.
Honestly, the more I tinker with Android, the more I'm terribly disappointed in Google. I mean, around Android 2 we were all excited by the potential of a first-class big-money supported open-source OS to really shake up the industry. It had so much potential.
Now? Well, it still has a lot of potential. Even Google seems kind of embarrassed by it, compared to the Chrome brand.
It does seem bizarre they don't make any ACL on external storage. I am sure this is not a strange isolated report. This problem must have been known for years both externally and internally.
On the side note, from old news I remember Google is indeed pushing forward with Chrome-brand, specifically Chrome OS.
Because people want to take their SD card, plug it in somewhere else, and have access to their photos / videos / songs. This necessitates using FAT. That is the case for most removable media.
WhatsApp is storing to external (on most devices FAT) storage (which was SD card on older devices, it's usually a separate directory/partition on newer ones) which does not have any ACL-like system due to FAT backwards compatibility.