Short Answer: Schisms in the church are the result of sin. However, God can use even the schisms to serve His purposes, by making a distinction between those who are walking according to His ways and those who are not.
First, it is important to distinguish between "the church" and "all the churches". (Compare 1 Timothy 3:15 and Romans 16:16 - same Greek word.)
In one sense, there is one church, comprised of all believers. Amongst believers, we should strive for unity, which means walking in humility, patience, tolerance, etc.
[I, Paul] implore you to walk in a manner worthy of the calling with which you have been called, with all humility and gentleness, with patience, showing tolerance for one another in love, being diligent to preserve the unity of the Spirit in the bond of peace. There is one body and one Spirit . . . one Lord, one faith, one baptism, one God and Father -Ephesians 4:1-6
The implication is that if there is not unity, it is due to sin. (See also Matthew 5:23-24.)
Believers gather together in local assemblies, which are referred to as "churches" in the Bible. For various reasons, local churches tend to take on unique personalities. (e.g. Revelation 2-3) However, the principle is the same - be of like mind. (2 Corinthians 13:11, Philippians 1:27, 2:2)
Now I exhort you, brethren, by the name of our Lord Jesus Christ, that you all agree and that there be no divisions among you, but that you be made complete in the same mind and in the same judgment. -1 Corinthians 1:10
Unfortunately, divisions are a reality. This wasn't something Martin Luther brought into the church... consider the following division between the ministries of Paul and Barnabas:
Barnabas wanted to take John, called Mark, along with them also. But Paul kept insisting that they should not take him along . . . And there occurred such a sharp disagreement that they separated from one another -Acts 15:37-39
The Root of the Problem
It is an unpopular message, but scripture teaches us clearly what the reason for division is. Divisions are the result of a person (or assembly) becoming worldly, devoid of the Spirit, following after their own lusts:
'In the last time there will be mockers, following after their own ungodly lusts.' These are the ones who cause divisions, worldly-minded, devoid of the Spirit. But you, beloved, building yourselves up on your most holy faith, praying in the Holy Spirit, keep yourselves in the love of God -Jude 1:18-21
In contrast to these wicked men, believers are instructed to walk by faith, pray in the Spirit, and remain in the love of God.
God doesn't desire divisions amongst His people, but if a person (or assembly) becomes worldly, lustful, etc. sometimes it is necessary for a separation to take place.
I wrote you in my letter not to associate with immoral people; I did not at all mean with the immoral people of this world . . . for then you would have to go out of the world. But actually, I wrote to you not to associate with any so-called brother if he is an immoral person . . . For what have I to do with judging outsiders? Do you not judge those who are within the church? . . . REMOVE THE WICKED MAN FROM AMONG YOURSELVES." -1 Corinthians 5:9-13
Of course, Luther believed the Catholic Church had become so corrupt in his day that he needed to separate from them. The Catholic Church responded with excommunication. In other words, the "two groups" felt they had irreconcilable differences in doctrine, and one or both sides were unwilling to continue in fellowship together.
But God can even work through situations caused by the evil motives of men, and work it all out for the good of His people (Genesis 50:20, Romans 8:28.) There is even a redemptive purpose which God can work through divisions - He can use the divisions to shed light on who is following in His ways and who isn't.
when you come together as a church, I hear that divisions exist among you . . . there must . . . be factions among you, so that those who are approved may become evident among you. -1 Corinthians 11:18-19