No, an attack is not a check.

The similarity between a check and an attack is in the way that rolls are handled, bonuses are added to the rolls and advantages/disadvantages are given. I believe the developers linked them together in order to not have to explain the same thing twice.

Throughout the document, the developers specifically mention checks AND attacks. When they talk about advantage/disadvantage, they point out that you can get this on a check, an attack roll or a saving throw.

Under each of the attributes, Checks, Attacks and Saving Throws are all outlined in their own manner. This trend continues through the playtest document. It would be redundant to do this if checks and attacks were the same.

The difference between ability checks, saving throws and attack rolls is clearly delineated in "Chapter 7: Using Ability Scores" of the Player's Basic Rules and Player's Handbook. Specifically,

The three main rolls of the game—the ability check, the saving throw, and the attack roll—rely on the six ability scores. The introduction describes the basic rule behind these rolls: roll a d20, add an ability modifier derived from one of the six ability scores, and compare the total to a target number.

This chapter [7] focuses on how to use ability checks and saving throws, covering the fundamental activities that creatures attempt in the game. Rules for attack rolls appear in chapter 9. (PBR, p. 57; PHB, p. 173)

Under the subsection Advantage and Disadvantage, it again indicates that these are separate rolls.

Sometimes a special ability or spell tells you that you have advantage or disadvantage on an ability check, a saving throw, or an attack roll. [Emphasis added] (Ibid.)

Attacks also are the only rolls that follow the natural 1 and natural 20 rules. All other rolls treat 1 and 20 just like any other number.