No, the Earth's axis isn't tilting wildly due to the ice caps melting.

I have received several emails about a website claiming that the Earth is tilting wildly from its usual 23.5° inclination, and that this is causing

If the change in season is due to the tilt of the earth, how come the polar ice isn't melting every season? - Quora

Melting Glaciers Have Shifted the Earth's Axis - GlacierHub

Earth has shifted on its axis due to melting of ice, study says - Bulletin of the Atomic Scientists

Hothouse” Earth? and Then There's Physics

Climate crisis has shifted the Earth's axis, study shows, Climate crisis

The earth tilts 23.5 degrees that makes seasons. If scientists were able to adjust the earth, what should be the tilt in degrees to have the most beneficial and comfortable season for

Melting ice shifted the Earth's axis, scientists discover - The Weather Network

What is the reason for the large ice sheets still present at the poles of our planet, when the last time there was a comparable amount was millions of years ago?

Earth Axis Tilt Change Caused by Melting Ice

Climate change is causing the Earth's axis to tilt faster than ever - Science

Melting ice sheets changing the way the Earth wobbles on its axis, says Nasa, Climate crisis

Glacial melting from climate change shifted Earth's poles

What will happen if the Earth's orbit, tilt on its axis, or rotation changes? Will that be catastrophic for life on Earth? - Quora