An excellent question! In the 16th and 17th centuries there were attempts by Protestant groups to reach out to the Christian East, but a few factors prevented any form of reunion from taking place:

Geo-Political Barriers

We must remember that at the time of the Protestant Reformation, the Christian East was under the control of a major Islamic empire. The ancient Patriarchates of Constantinople, Antioch, Jerusalem, and Alexandria were all safely within the borders of the Ottoman Empire. The Ottomans were, for the most part, content to let the Christians govern themselves but this doesn't mean there was anything close to what we would consider freedom of religion and separation of church and state. The Ottomans had great influence in determining who would be the next Patriarch of Constantinople, and several troublesome Patriarchs met an untimely demise. Likewise, in Western Europe, it was not as simple as just "changing sides." France, Germany, and the low countries descended into warfare and chaos. The French Wars of Relgion, the Dutch Revolt, and the Thirty Years War in Germany all kept Western Europe preoccupied. The Protestant Reformation divided entire nations. For the various Protestant factions to "become Orthodox" would have necessarily required entire national governments to shift their allegiance: something that just wasn't feasible in France and Germany. That being said, there certainly were attempts by Lutherans and Calvinists to reconcile to the Christian East. Two different attempts were made under Ecumenical Patriarch Jeremias II Tranos (1572-1595) and Ecumenical Patriarch Cyril Lucaris (1620-1638). Both attempts ultimately met with failure.

Ecumenical Patriarch Jeremias II Tranos and the Tubingen Theologians

In 1575 Lutheran theologians Jakob Andreae (notably one of the authors of the Formula of Concord) and Martin Crusius sent a letter to Patriarch Jeremias II, along with a Greek translation of the Augsburg Confession. In 1576, Jeremias II responded with a scathing rebuttal in which he rejected nearly every point of the confession. Andreae and Crusius sent a defense in 1577, which was met with similar rebuttal, and communication broke down thereafter. This incident tells us a lot about what both sides hoped to accomplish. Jeremias II maintained that the Christian East had no desire for reformation. That the Lutherans did not embrace the censure of the Ecumenical Patriarch shows us that they had no desire to bring themselves under the authority of the Christian East. The Lutherans sought more than just the rejection of Rome: they really believed in their own theological system, a system that was plainly incompatible with the teachings of the Eastern Church.

Cyril Lucaris and His Enigmatic Confession

Another incident further shows us the extent to which the Protestant West and Eastern Church were simply unwilling to accept one another. In 1629, a mysterious document appeared claiming to have been authored by none other than Cyril Lucaris, Patriarch of Constantinople, which claimed to be a confession of the faith held by the Eastern Church. The document affirms highly Calvinistic doctrines: justification by faith alone, predestination, and the rejection of any distinction between bishops and priests. The document caused no shortage of scandal, and even today its authorship is highly contested. Patriarch Lucaris did have notably cordial relations with Protestant ambassadors in Constantinople. In his youth he had studied in Venice and Padua. He helped lead the opposition against a proposed union of the Roman and Ukrainian Churches at the Synod of Brest-Litovsk in 1596, and despite his earlier studies was highly critical of Roman Catholicism, especially the Jesuit order. Yet, outside of the Confession attributed to him, there is no record of him supporting Calvinist thought.

The document itself is also highly questionable: it was published in Latin, in the Calvinist city of Geneva. It seems odd that the staunchly anti-Latin Greek Patriarch would publish a document in Latin, in Geneva, which claimed to represent the position of the Eastern Church. The authorship of this confession continues to be debated today by Calvinist and Eastern Orthodox polemicists seeking to promote their own side. What is not debated is the Eastern Church's overall response: the Confession of Cyril Lucaris was condemned at several synods over the next century. At the Synod of Jerusalem in 1672, Patriarch Dositheus II of Jerusalem submitted a point by point rebuttal of Lucaris' (alleged) Confession, which was subsequently approved by the Synod. The Confession of Dositheus leaves no room for compromise. The document not only rejects Calvinism, but affirms several Roman Catholic teachings that the Orthodox Church is usually wary of, most notably the doctrine of transubstantiation regarding the Eucharist. This can be attributed to the fact that the Eastern Church frequently sent clergy to be trained in Italy during the time of the Ottoman Empire.

Theological Barriers and Conclusion

What the above two events demonstrate quite plainly is that Protestants were quite committed to their own theological system, a system that the Eastern Church simply did not accept and was unwilling to negotiate on. Protestant slogans like sola fide and sola scriptura had no place in the Christian East. Many Protestants, for their part, were still highly committed to distinctly Western theological concepts. Lutherans, Calvinists, and Anglicans all kept the filioque phrase in their versions of the Niceno-Constantinopolitan Creed, the inclusion of which had been the immediate the Great East-West Schism in the first place. Likewise, the Eastern Orthodox Church practiced many things the Protestants would never accept: the veneration of saints (especially through the use of icons), prayers to Mary, Apostolic Succession, etc. With the exception of certain Anglo-Catholic groups, those barriers remain to this day.

Addendum

Since this comment is gaining traction, I have a few more thoughts to share. We need to be careful to remember that the Lutheran Reformation was not explicitly separatist at the beginning. The explicit goal of Luther and his followers was to reform the Western Church, not break from it. Even as late as the Council of Trent many Lutherans still hoped that religious unity could be restored. When the Reformation began both sides very much considered it an internal matter for the Western Church to resolve. It was only after several decades, during which Protestants and Catholics took increasingly extreme steps, that the religious makeup of Europe consisted of distinct Protestant churches standing against a united Roman Catholic Church.

In our modern, secularized, and pluralistic society the question of "why didn't they just convert" seems sensible. However, such an idea would have been completely alien to the average person, even clergy, living in early modern Europe. Religious affiliation and structure was largely left to major political and religious leaders to decide. Although idealized as a populist movement, Protestantism only succeeded because the political powers of Western Europe fought for it. In the Holy Roman Empire especially, the Protestant movement was bolstered by the defection of princes and electors who desired more sovereignty and independence from a religious hierarchy controlled in Italy. It is unlikely they would be willing to place themselves under another foreign hierarchy.

Hey there,

Just to let you know, your question is fine, and we're letting it stand. However, you should be aware that questions framed as 'Why didn't X do Y' relatively often don't get an answer that meets our standards (in our experience as moderators). There are a few reasons for this. Firstly, it often can be difficult to prove the counterfactual: historians know much more about what happened than what might have happened. Secondly, 'why didn't X do Y' questions are sometimes phrased in an ahistorical way. It's worth remembering that people in the past couldn't see into the future, and they generally didn't have all the information we now have about their situations; things that look obvious now didn't necessarily look that way at the time.

If you end up not getting a response after a day or two, consider asking a new question focusing instead on why what happened did happen (rather than why what didn't happen didn't happen) - this kind of question is more likely to get a response in our experience. Hope this helps!