That is absolutely not true. When I read it, which I have read it a few times, I don't remember any where where it says that the Earth is flat in a literal meaning. Even when Jesus said go to the ends of the earth to spread the gospel, it did not mean that there was an end of the earth. Instead it was figurative language that suggests more of a universal scope and not the geographical goal.
In Acts 1:8, Jesus said to the disciples, “But you will receive power when the Holy Spirit comes on you; and you will be my witnesses in Jerusalem, and in all Judea and Samaria, and to the ends of the earth.” (NIV)
In the current round earth model the sun is the center of our solar system and the earth is spinning every 24 hours and also orbiting around the sun. The earth is in motion. Any bible verse which says the earth or the world is not moving supports the flat earth model in which the flat earth does not move at all and everything rotates around it. In addition our solar system is flying through space as well.
Verses that say the earth can't or does not move:
1 Chronicles 16:30:
"The world is firmly established; it cannot be moved."
Psalm 104:5
"He set the earth on its foundations; it can never be moved."
Psalm 93:1
"The world was set in place; it cannot be moved."
Psalm 96:10
"The earth stands firm; it cannot be moved."
If God set the earth on its foundations and it can NEVER be moved then how is the earth spinning to create the round earth day/night cycle?
If God set the earth on its foundations and it can NEVER be moved then how is the earth orbiting the sun?
If God set the earth on its foundations and it can NEVER be moved then how is solar system (with the earth in it) moving through space?
If you believe that the earth is not moving then none of those things listed above are true and you would have no choice but to follow a more biblical flat earth model.
This verse the Devil shows Jesus all the kingdoms of the earth from the top of a mountain. Since people lived all over the earth (like the Aztecs in South America) the only way for Jesus to see all the kingdoms of the earth from the top of a mountain was if the earth was flat:
Matthew 4:8