A similar (and for our purposes equivalent) claim is that God cannot stand the presence of sin, and therefore the only people who can be welcome in heaven are those who are free of any sin. In other words, those who have "fallen short of the glory of God" are not welcome in the presence of God.
Here is a short proof that this belief contradicts scripture.
Jesus says that only God is good. (Mark 10:18, Luke 18:19)
If only God is good, then the angels must also fall short of "God's glory" (c.f., Romans 3:23). According to the modern Protestant understanding of sin, this makes the angels sinners. [This understanding of sin is flawed, but that is another story. Catholics have a different translation of Romans 3:23 that shines a very different light on the topic.]
If only the perfect can be in God's presence, then the angels must also be forbidden in heaven. In fact, their plight is worse than humanity's because they do not participate in the mercy shown to humankind owing to Christ's work [Hebrews 2:16]
Obviously, angels are allowed in God's presence, so we reach a contradiction. QED.
One interesting point about the above is that the most obvious way to "fix" it is to try to better understand what makes the angel's ungoodness/imperfection/sin/falling-short-of-God's-Glory different from the same thing in humanity. At first it doesn't seem hard to claim that somehow humans sin in a qualitatively different way than angels do... and that is certainly true! But then one has to drill down on the subject of what is "sin" and what sin "counts" with respect to the topic of God's judgment and/or heaven [a question up for debate according to Romans 5:13]...and the more one tries to clarify and make distinctions there, the harder the rest of the Evangelical case becomes.... which is a subject for a different day.
Commentary on the Garden of Eden
This belief, that God judges using a standard of perfection and that God can only countenance those who are perfect, has no basis in scripture. Many people point to the story of the Garden of Eden as proof that a single failing casts you out of God's presence. However, there are 3 very powerful rebuttals to this, any one of them would be sufficient on its own to dismiss that claim:
- Not once in any gospel or any of the evangelism shown in Acts is the Garden of Eden mentioned. If a core principle of Christian salvation relates to the sin of Adam and Eve in a singular way, you would expect the gospels and the book of Acts to talk about the Garden as much as modern Evangelicals do, but in reality NONE of the gospel writers even reference it and it shows up nowhere in Acts, which is the principal showcase of Apostolic Evangelism.
- Contrary to what most people are taught, the explicit reason why Adam and Eve were kicked from the Garden is not because of their status of sinners. The reason God removed them from the Garden, as described explicitly in Genesis 3:22, is that God did not want them to become like the heavenly host: who are both immortal and know the difference between good and evil. Note that God says "they will be like us" if they eat the fruit from the tree of life. Thus, even the proclamation of eviction was based on humanity's coming one step closer to being like the heavenly host, not the opposite.
- While it is true that the "fall of Adam" did lead to a spiritual death for humanity, one cannot claim that this spiritual death sufficed to prevent communion with God or entry into God's realm. The prophets were able to commune with God even though they were Adam's descendants, and Enoch and Elijah are taken up to God without ever dying (and before the death of Christ). We are told explicitly that "Enoch walked with God" and did not die. What could this possibly mean if it does not refer to his early entry into God's eternal presence? Similarly, Elijah is taken up to heaven in a whirlwind.