Does a Grapple or Shove count as a "melee attack" for the third benefit of the Mobile feat?

Question

The third benefit of the Mobile feat (PHB, p. 168) enables you to not provoke opportunity attacks by creatures you try to make a melee attack against.

Does it still apply if the attack you made was a Grapple or Shove?

Answer 1

Yes, grapples and shoves count as melee attacks for the Mobile feat, according to RAW

The third benefit of the Mobile feat says (PHB, p. 168; emphasis mine):

When you make a melee attack against a creature, you don’t provoke opportunity attacks from that creature for the rest of the turn, whether you hit or not.

Thus, it only triggers this effect on a melee attack and does not care if that melee attack results in a hit or not.

Grappling and shoving are melee attacks

The rules for grappling say:

When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. 

The rules for shoving say:

Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you.

Rules designer Jeremy Crawford has also made an unofficial ruling to the same effect in this tweet:

An attack involves an attack roll or doing something that the rules call an attack, like grappling or shoving.

So, grappling and shoving are are defined as attacks in the rules and designer clarification has verified that (many times in fact). Thus, they qualify as melee attacks for this part of the mobile feat.

Grapples/Shoves cannot hit, but this technically does not disqualify them

We know that grappling and shoving are attacks. However, they are unusual attacks because they do not use an attack roll.¹ And because of that they cannot hit or miss - only succeed or fail (see Does grappling count as a hit?).

This is important because Mobile says that the effect triggers "whether you hit or not".

Despite the fact that they cannot hit or miss, technically the wording still leaves room for grapples/shoves to qualify. If it had said "whether you hit or miss" then you would not be able to use a grapple. However, it does not say that, it says "hit or not" and technically a grapple attack will always not hit because it cannot hit (it can succeed or fail). Thus, they still work via a strict RAW reading of the language.²

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¹ Because there is some confusion on the matter, the fact that shoving/grappling does not involve an attack roll does not in any way make them not an attack. Most attacks involve an attack roll; grapples and shoves are unusual in the fact that they don't involve an attack roll, but they are still attacks. As Jeremy Crawford once again clarified in an unofficial tweet:

The grapple option in the Player's Handbook is an attack, but it uses an ability check in place of an attack roll.

² This kind of weirdness in the language seems to indicate that the feat was written without consideration for the fact that melee attacks that do not involve attack rolls do exist (they are in fact extremely rare). This seems more likely than an attempt to specifically exclude grapples/shoves. In any case, the language does not actually exclude them so my logic stands regardless of intent.

Answer 2

No, it has to be an attack roll

In 5e, "an attack" has a special meaning — it means something with an attack roll. When you throw a fireball to an enemy, you attack them, but it is not "an attack" in terms of 5e. See What counts as an attack? :

If there's ever any question whether something you're doing counts as an attack, the rule is simple: if you're making an attack roll, you're making an attack
(PH page 194)

Despite Grapple and Shove are called "special melee attacks" in PH, in terms of 5e they are actually contests:

Battle often involves pitting your prowess against that of your foe. Such a challenge is represented by a contest. This section includes the most common contests that require an actton in combat: grappling and shoving a creature. The DM can use these contests as models for improvising others.
(PH, page 195)

They are called "special melee attacks" within a particular context — is possible to make them as a part of an Attack action with multiple attacks:

you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them

More info: What does upper-case-A-Attack action vs. lower-case-a-attack mean?

The feat description assumes it has to be an attack that can hit:

When you make a melee attack against a creature, you don’t provoke opportunity attacks from that creature for the rest of the turn, whether you hit or not.

Grapple and Shove cannot hit, they are not "attacks" in terms of 5e, hence, cannot trigger the Mobile feat.