The short answer to your question is no, the apostle John was not teaching that God’s wrath has been satisfied toward both the elect and non-elect. Since the word propitiation means "appeased," we can be certain John was only talking about those who would believe in 1 John 2:2.

There is a lengthy, but extremely helpful footnote on that passage in the MacArthur Study Bible. With simplicity and clarity, John explains what the apostle meant by the phrase “the whole world.” Take a look:

for the whole world This is a generic term, referring not to every single individual, but to mankind in general. Christ actually paid the penalty only for those who would repent and believe. A number of Scriptures indicate that Christ died for the world (John 1:29; 3:16; 6:51; 1 Tim. 2:6; Heb. 2:9). Most of the world will be eternally condemned to hell to pay for their own sins, so they could not have been paid for by Christ. The passages which speak of Christ’s dying for the whole world must be understood to refer to mankind in general (as in Titus 2:3,4). “World” indicates the sphere, the beings toward whom God seeks reconciliation and has provided propitiation. God has mitigated His wrath on sinners temporarily, by letting them live and enjoy earthly life (see note on 1 Tim. 4:10). In that sense, Christ has provided a brief, temporal propitiation for the whole world. But He actually satisfied fully the wrath of God eternally only for the elect who believe. Christ’s death in itself had unlimited and infinite value because He is Holy God. Thus His sacrifice was sufficient to pay the penalty for all the sins of all whom God brings to faith. But the actual satisfaction and atonement was made only for those who believe (cf. John 10:11,15; 17:9,20; Acts 20:28; Rom. 8:32,37; Eph. 5:25). The pardon for sin is offered to the whole world, but received only by those who believe (cf. 4:9,14; John 5:24). There is no other way to be reconciled to God. (The MacArthur Study Bible, 1 Jn. 2:2)

Concerning God’s wrath being unsatisfied toward unbelievers, Paul said this in Romans 2:5, But because of your stubbornness and unrepentant heart you are storing up wrath for yourself in the day of wrath and revelation of the righteous judgment of God, who will render to each person according to his deeds.

Also see chapters 6-19 of Revelation, which chronicle the future judgments of God. That section is characterized by one wave of wrath after another. 6:16-17 says, And they said to the mountains and to the rocks, “Fall on us and hide us from the presence of Him who sits on the throne, and from the wrath of the Lamb; for the great day of their wrath has come, and who is able to stand?”

I hope that helps answer your question, Caleb. Thanks for your comment.