I do not believe that the wrath of God was satisfied on the cross because:
1) I cannot find such a sentence in the Bible (please correct me if I am wrong in this).
2) I can only notice that the prooftexts for such doctrine (such as Romans 5:9) refer to God’s future judgment.
3) God already loved the world before He sent His Son to die for us on the cross and that was why He sent Him (John 3:16).
4) God has wrath afterwards (Acts 5:5,10; Romans 2:5, 8-9; Revelation 16).
It seems to me that also the apostle Paul did not believe that God’s wrath was satisfied on the cross, when he wrote afterwards: “Give place unto wrath: for it is written, Vengeance is mine; I will repay, saith the Lord.” (Romans 12:19b, KJV). If Christ’s death ‘satisfied the wrath of God’, as those who believe in the Satisfaction Theory of the atonement claim, then the sin of unbelief would also have been paid for so that they could “deny the Lord that bought them” without “bringing upon themselves swift destruction” (contrary to 1 Peter 2:1).
I think it is proper to conclude with a note by one of the early Church’s fathers, Gregory of Nazianzus:
“Is it not plain that the Father received the ransom, not because He Himself required or needed it, but for the sake of the Divine government of the universe (…)?” 
G. Frederick Wright put it thusly, rephrasing and quoting the words of Charles G. Finney: ‘In further defining justification, Finney tries to distinguish between the ground of justification and the conditions of its exercise, maintaining that the atonement of Christ was not the ground, but simply one of the conditions, of justification. The ground (by which he means the moving, procuring cause of justification, that in which the plan of redemption originated as its source, and which was the fundamental reason or ground of the whole movement) “was the benevolence and merciful disposition of the whole Godhead, Father, Son, and Holy Spirit. This love made the atonement, but the atonement did not beget this love. The Godhead desired to save sinners, but could not safely do so without danger to the universe, unless something was done to satisfy public, not retributive justice. The atonement was resorted to as a means of reconciling forgiveness with the wholesome administration of justice.” 
 The same Idea is found in The Atonement of Jesus Christ by William Booth:
‘If Christ has satisfied all the claims which the Law has upon those who transgressed it, He must have satisfied also the claim involved in the unbelief entertained down to the last moment of life. Consequently, “if the debt is paid, the obligation is discharged, and the debtor is free.” It would follow inevitably that, whether I believe the good tidings or not, my unbelief cannot affect the fact; and whatever wickedness may be involved in my refusing to believe, that wickedness itself is also paid for if all my debt is discharged – I am free [to do what I want]’.
– William Booth, The Atonement of Jesus Christ (reprinted 2015 version), p. 5.
This is a republished version of the 1922 edition (you can get it here).
 As stated in William Greenough Thayer Shedd, A History of Christian Doctrine: Volume II (New York: Charles Scribner’s Sons, 1868), p. 245. Saint Gregory of Nazianzus lived 330-390 CE.
 As quoted from G. Frederick Wright, A Biography of Charles Grandison Finney (Ohio: Oberlin College, 1891), p. 137. The reference to the quote is Finney’s Theology, p. 550.