On the Melting of the Arctic Ice I have read that:
Polar ice caps are melting as global warming causes climate change. We lose Arctic sea ice at a rate of almost 13% per decade, and over the past 30 years, the oldest and thickest ice in the Arctic has declined by a stunning 95%. If emissions continue to rise unchecked, the Arctic could be ice-free in the summer by 2040. https://www.worldwildlife.org/pages/six-ways-loss-of-arctic-ice-impacts-everyone
QUESTION: If the Arctic ever becomes ice-free, what percentage of the earth's current landmass would then be under water?
 A: The ice of the Arctic floats on water. Melting it would make no difference to sea level.
You can test this concept yourself. Half-fill a large glass jar with water. Add some ice, making sure it all floats and does not touch bottom. Note the level of the water after the ice has been added. Now wait until the ice melts, and you will see that the water level does not change.
The reason for this is that the density of ice is lower than that of water - that's why ice floats. In floating it displaces a water mass that's equal to its own mass. Once it melts, that mass just fills the volume of water that was displaced.
None of this applies to ice that rests on land, e.g. Greenland, a large portion (not all) of Antarctica, glaciers, etc.
A: As the other respective persons already pointed out the reasons of rising sea level briefly, I am just highlighting the points.

Melting of floating icebergs doesn't rise sea level.


Sea level increases because of melting of icebergs that are resting on land.


A significant rise of sea level is caused due to the thermal expansion of water for global warming.


Question: If the Arctic ever becomes ice-free, what percentage of the earth's current landmass would then be under water?

There are around 30M km³ of land ice in Antarctica, and around 10% of that in Greenland, which basically just makes up for the 9% shrinkage on melting. There is also a lot of ice in the Arctic region (In comparison, there is very small amount of land ice) . If all of the ice is melted, the global sea level goes up by around 70m.
For this rising sea level, the US east and gulf coast, especially Florida, the British Isles, Denmark, Holland, North Africa, a lot of Russia, India, Pakistan, Bangladesh and a lot of SE Asia and coastal China will be underwater. Here are some maps for better understanding:



So these are the places. But that is about it, plenty of land, corresponding to the present continents, is still left. Specifically African continent is nearly untouched. Hence, there is a lot of land for human and other land living animals to live.
But you should notice that the parts of the Earth that would be PERMANENTLY underwater are already underwater. We are nearing the expected end to the Late Cenozoic Ice Age that started about 40 to 50 million years ago. We expect about 110 million years of no ice as is what the 150 million years ice age cycle predicts. Then we are expected to enter another ice age cycle of about 40 million years.

We are just lucky to be living in the only time in over 400 million years where sea levels have been so stable. The diagram shows up as a level line on the scale needed to display sea level change over that time. Notice there are only a few level parts of the graph of sea level change in the blue graph below and they are all in recent times. Before recent times sea levels were either rising extremely fast or falling fast. They were never stable like they are today.

