No. "Unarmed strike" has a specific meaning in the rules.
The phrase "unarmed strike" is something called a "game term"1, that is, a word or phrase that has a special meaning in the context of the game rules that is different from its plain english meaning. In this case, an unarmed strike is defined so:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
So when the Martial Arts ability says "unarmed strike" it is referring specifically to this game mechanic. Grapples and shoves are totally separate mechanics, and while they may be able to be referred as unarmed strikes in the plain English sense of the phrase, they do not count as unarmed strikes for the purpose of rules that refer to unarmed strikes.
1 "Game term" is general hobby lingo in this case, the D&D 5e rules do not actually define or use the phrase, but nevertheless use the concept extensively. Contrast with D&D 4e which had a "keyword" system for this.