That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.
SRD, page 27 - Fighter Extra Attack:
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
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. If you’re able to make multiple attacks with the Attack action, this attack replaces one of them.
Is there any indication or rule provided for if the text (one weapon attack only) in Haste is specifying what the Attack Action provides or if this is a limitation placed on the Attack Action?
The former means that shoves, grapples, and any other rule or feature that replaces Attacks would not be valid with the 2nd attack provided by Extra Attack or Haste's Attack Action at all.
The latter means that it is valid to make one weapon attack, then use the other attack from Extra Attack to do anything else available to a creature that is not a weapon attack.