It depends on what you mean by "destroy the earth". If your friend was referring to destroying the actual planet so that it no longer exists then quite the opposite is true (see 2 Peter 3 below). If your friend was referring to destroying its surface and all life on it then yes God does say as much.
In Genesis 8, right after God destroyed almost every living thing with a global flood, He says:
21 The Lord smelled the soothing aroma; and the Lord said to Himself, “I will never again curse the ground on account of man, for the intent of man’s heart is evil from his youth; and I will never again destroy every living thing, as I have done.
22 “While the earth remains,
Seedtime and harvest,
And cold and heat,
And summer and winter,
And day and night
Shall not cease.” (NASB)
So here you have God saying that He wouldn't destroy all life and the world systems (seasons, days & nights, ...). He is more specific in Genesis 9:11 where He says:
I establish My covenant with you; and all flesh shall never again be cut off by the water of the flood, neither shall there again be a flood to destroy the earth. (NASB)
Here God is specific that He wont use a flood to cut off all life and He even refers to this kind of catastrophic event as something that would "destroy the earth" though a flood would obviously leave the planet intact.
God does say that He will eventually destroy the planet though. In 2 Peter 3:10 it says:
But the day of the Lord will come like a thief, in which the heavens will pass away with a roar and the elements will be destroyed with intense heat, and the earth and its works will be burned up. (NASB)
And again in Revelation 21:1 it says:
Then I saw a new heaven and a new earth; for the first heaven and the first earth passed away, and there is no longer any sea. (NASB)
Now the scope of God's promise to not destroy the earth in Genesis 8-9 is debatable. In its broadest sense He could be saying that "while the earth remains" He wont destroy all life. In the narrowest sense He could be saying that He wont destroy the earth with a flood. Neither contradict God's later statements in 2 Peter 3 and Revelation 21. If it is the former then the clauses "as I have done" and "while the earth remains" are important because the coming destruction involves the earth not remaining. If it is the later then the coming destruction is by fire and thus not excluded.
In summary, it does appear that God promised to not destroy the earth in the sense that He did with the flood (destroy all life). But He does promise to destroy the earth in that it will be no more.