I am unaware of any official objections published by a Protestant denomination, but as a result of looking into Ignatian Spirituality, a couple of things give me cause for concern. Although the objectives of the spiritual exercises are praiseworthy, an examination of the history behind this revival is useful.
Ignatius of Loyola was a 16th-century Spanish priest, theologian, and founder of the Society of Jesus (Jesuits). While the life and devotion of Ignatius of Loyola was truly remarkable, his vows—still taken by all Jesuits—include “absolute obedience to the Pope.” Whilst nobody is suggesting Protestants who engage in these spiritual exercises will be compelled to take any such vow, they might want to do some research into this organisation. After presenting a brief overview of the history of the Jesuits, the article below makes this comment:
The Jesuits are still active in the world today, though the military actions of those early years have been left behind. The goal of spreading the Catholic faith is still their primary objective, and they do it through missionary work and education.
Another practice is deep and constant meditation on the sins that have been committed, so as to rouse intense sorrow for sins... Source: https://www.gotquestions.org/Jesuits-Society-of-Jesus.html
As the Wikipedia article you quote from says, the first week of contemplation is devoted to sin and God’s mercy. Whilst it is essential that sinners must face up to and repent of their sins, our focus should be on God’s plan of salvation and how salvation is entirely the work of God, from start to finish. Protestants believe no amount of good works, no adherence to the traditions and laws of men, can add to the salvation that is granted freely to all who come to saving faith in Christ Jesus.
This is possibly one of the most fundamental differences between Catholics and Protestants. I do not say this to provoke an argument, but merely to express one of my concerns about engaging in these spiritual exercises.
What really concerns me, though, is this paragraph in the Wikipedia article you quote from and, in particular, the last sentence (emphasis mine):
"Discernment" is very important to Ignatian thought. Through the process of discernment, the believer is led toward a direct connection between one's thought and action and the grace of God. As such, discernment can be considered a movement toward mystical union with God, and it emphasizes the mystical experience of the believer. This aspect of the Spiritual Exercises reflects the trend toward mysticism in Catholic thought which flourished during the time of the counter-reformation (e.g., with Teresa of Ávila, Francis de Sales, and Pierre de Bérulle). However, while discernment can be understood as a mystical path, it can also more prosaically be understood as a method of subjective ethical thought. The Exercises emphasize the role of one's own mental faculties in deciding what is right and wrong. https://en.wikipedia.org/wiki/Spiritual_Exercises_of_Ignatius_of_Loyola#Spiritual_vi
As a Protestant that sounds a very loud alarm bell, spiritually speaking. Protestants rely on the word of God in deciding what is right and wrong:
Do not let this Book of the Law depart from your mouth; meditate on it day and night, so that you may be careful to do everything written in it” (Joshua 1:8).
Blessed is the man who does not walk in the counsel of the wicked or stand in the way of sinners or sit in the seat of mockers. But his delight is in the law of the Lord, and on his law he meditates day and night (Psalm 1:1-2).
Bible-believing Christians are grounded in God’s word, and focus on what God, in Christ Jesus, has done to save us from our sinful condition. Our mental faculties are not reliable and we can be easily deceived. 1 Corinthians 2:10-16 describes how the Spirit of God alone knows the thoughts of God:
This is what we speak, not in words taught us by human wisdom but in words taught by the Spirit, expressing spiritual truths in spiritual words... But we have the mind of Christ.
The point is that Christians do not rely on their own mental faculties in deciding what is right and wrong. Christians rely on the words taught by the Holy Spirit, which are found in the Holy Bible. Born-again Christians receive the Spirit by believing what they have heard about the gospel and are warned against trying to attain their goal by human effort. We are justified before God by faith in Christ Jesus (Galatians 3:3-4; 11). We follow Christ, and do not place our faith in methods designed by humans, regardless of how holy and sincere they are.
Everything necessary for our salvation has already been accomplished by Christ Jesus. Christians who are indwelt by the Holy Spirit model their lives on Him.
These views and concerns are personal, although I am a Christian of the Reformed Protestant persuasion.