Though the Christian writers knew the concept of a new earth from Isaiah's inspired writings, Peter and John both were inspired to expand on what Isaiah informed us.  Peter's writing actually impacts directly on this subject. It is immensely informative to meditate on his words.

2 Peter 3:5-7 For this they, willfully, forget—that there were, heavens, from of old, and, an earth, on account of water and by means of water, compacted, by God’s word,— 6By which means, the world that then was, with water being flooded, perished; 7While, the heavens and the earth that now are, by the same word, have been stored with fire, being kept unto the day of judgment and destruction of the ungodly men. (Rotherham)

(KJV) 5 For this they willingly are ignorant of, that by the word of God the heavens were of old, and the earth standing out of the water and in the water: 6Whereby the world that then was, being overflowed with water, perished: 7But the heavens and the earth, which are now, by the same word are kept in store, reserved unto fire against the day of judgment and perdition of ungodly men.

In verse five and verse seven, we have the previous heavens and earth opposed to the present heavens and earth, "which are now" (KJV). This is the key to understanding what the new earth and the new heavens mean.

We need to ask ourselves, did the actual antediluvian physical heavens and earth perish? Peter tells us that the antediluvian earth and heavens perished! If you argue that he used the word "world" then you are ignoring his following statement –"But the heavens and the earth, which are now!" Thus we can understand that he referred to the previous earth, society of men, and the previous heavens, the ruling system in place then.

II Peter 3:10 But the day of the Lord will come as a thief in the night; in the which the heavens shall pass away with a great noise, and the elements shall melt with fervent heat, the earth also and the works that are therein shall be burned up.

13Nevertheless we, according to his promise, look for new heavens and a new earth, wherein dwelleth righteousness. (KJV)

If we apply the knowledge gained about II Peter 3:5-7, we then understand that this new heavens and new earth must as before, to be consistent, refer to a new society of men and a new system of government.

These prophecies and many, many more all show the Kingdom rule being installed on this earth with dominion over the nations. Psalms says " he shall continue as long as the sun, and before the moon forever." Clearly this heaven is old, and getting older forever! So if the physical heavens remain unaffected, likewise in harmony with the above direct points, the physical earth shall remain forever as even Ecclesiastes 1:4 emphasized: "A generation goes, and a generation comes; but the earth abides forever."

Another indirect scripture, in harmony with the evidence above, is the one in which no letter of any prophecy shall remain unfulfilled. It was God's will for the present earth to become a Paradise, and so it shall become since God may not be found a liar to be. (Matt 5:18, Luke 16:17, Titus 1:2)

(KJV) Zechariah 9:9-10, " Rejoice greatly, O daughter of Zion; shout, O daughter of Jerusalem: behold, thy King cometh unto thee: he is just, and having salvation; lowly, and riding upon an ass, and upon a colt the foal of an ass.*n9.1 10And I will cut off the chariot from Ephraim, and the horse from Jerusalem, and the battle bow shall be cut off: and he shall speak peace unto the heathen: and his dominion shall be from sea even to sea, and from the river even to the ends of the earth.

Psalms 72:1-7 O God, give Your judgment to the king, and Your righteousness to the king's son; 2that he may judge Your people with righteousness, and Your poor with judgment. 3Let the mountains and the hills raise peace to Your people. 4He shall judge the poor of the people in righteousness, and save the children of the needy; and shall bring low the false accuser. 5And He shall continue as long as the sun, and before the moon forever. 6He shall come down as rain upon a fleece, and as drops falling upon the earth. 7In His days shall righteousness spring up; and abundance of peace till the moon be removed. 8And He shall have dominion from sea to sea, and from the River to the ends of the earth. (AB)

Daniel 7:13-18 One like the Son of Man coming with the clouds of heaven, and He came to the Ancient of Days, and was brought near to Him. 14And to Him was given the dominion, and the honor, and the kingdom; and all nations, tribes, and languages, shall serve Him. His dominion is an everlasting dominion, which shall not pass away, and His kingdom shall not be destroyed. 15. . . 17These four beasts are four kingdoms that shall rise up on the earth, 18which shall be taken away; and the saints of the Most High shall take the kingdom, and possess it forever and ever. (AB)

Given the above points and scriptures, the scriptures demonstrate Christ's kingdom to be "from sea to sea, to the ends of the earth." We may therefore conclude that the sea in Rev 21:1 refers not to the physical sea, but rather refers to the sea of mankind in which Satan, the original serpent, swims.  Perhaps one might also define it as the unruly part of mankind, since the sea never is still. (Refer to the scriptures below for proof.) This gives eminent sense to the scriptures in Revelation and other places where the beast come out of the sea and where the big snake living in the sea is being destroyed.

Revelation 21:1 And I saw a new heaven and a new earth: for the first heaven and the first earth are passed away; and the sea is no more. (ASV)

Isaiah 57:20 But the wicked are like the raging sea, which cannot rest, and the waves thereof cast up dirt and mire. (DRC)

Revelation 17:15 And he saith to me, `The waters that thou didst see, where the whore doth sit, are peoples, and multitudes, and nations, and tongues;' (YLT)

Daniel 7:3, 23 And four great beasts came up out of the sea, each differing from one another. 23And he said, The fourth beast shall be the fourth kingdom on the earth, which shall excel beyond all other kingdoms, and shall devour the whole earth, and trample and destroy it. (AB)

Job 26:13 He has calmed the sea with His might, and by His wisdom the whale has been overthrown. 13And the barriers of heaven fear Him, and by a command He has slain the apostate dragon. (AB)
(DRC) 12By his power the seas are suddenly gathered together, and his wisdom has struck the proud one. 13His spirit hath adorned the heavens, and his obstetric hand brought forth the winding serpent.

The scriptures are then seen to support the interpretation of the sea in Revelation and other places were appropriate, as the sea of wicked mankind that cannot find rest.

Ephesians 6:12 because we have not the wrestling with blood and flesh, but with the principalities, with the authorities, with the world-rulers of the darkness of this age, with the spiritual things of the evil in the heavenly places; (YLT)

As seen here, Christians wrestle with this old world's heavenly forces; this is the old heavens that shall be replaced when Satan and his demons are abyssed.

If the above is sufficient proof for all is a good question, but at the moment that is all there is.

It would be exaggerating things to claim that this dogma should be considered part of the very rock foundation dogma that affects a person's chances for salvation!  Nonetheless, it surely belongs to a second level (so to say) foundation doctrine. It is an important part of a Christian's all around knowledge regarding his faith and basic understanding of the scriptures.