In MSA we can say لا تتدخّل في شؤون الاخرين or لا تتدخّل في شؤون الغير in the imperative addressed to a male. The verb form will change for other subjects/moods/tenses; for the third person past we will have لم يتدخّل.
There's probably a more casual expression too but I don't know of one.
Naturally; there are thousand an more ways of saying it.
Here are some of them:
"Mind your own business !"
لا تحشر أنفك في ما لا يخصك!
إبتعد عن ما لا يعنيك!
As for "He is minding his own business" same thing applies..
هو يتجنب عن ما لا يخصه ه
هو يغض نظره عن ما لا يعنيه ه
Thank you both. As regards "He was minding his own business," the meaning can vary in English, but the one that I have in mind is "He was thinking his own thoughts, and not interfering with anyone else."
Might there be an idiomatic Arabic expression conveying something such as that?
I see that the Reverso website gives for "minding his own business" يتدبر عمله الخاص .
I am not sure about that, I have got a feeling that it is not really an idiomatic Arabic expression, but am hoping that one of the WR community that have been so helpful to me over the years can come to my rescue !
يتدبر عمله الخاص sounds like "He manages his own work/job " to me ,
about "He was minding his own business" i would suggest ;كان يهتم بشؤونه الخاصه
But the meaning changes a little bit with this detail you added : "He was thinking his own thoughts, and not interfering with anyone else."
my understanding and attempt is : كان مستقلا في تفكيره وقراراته , literally in English : He was independent in his thoughts and decisions