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

Melting glaciers due to climate change caused Earth's axis to shift since mid-90s-Tech News , Firstpost

More than 40% of Antarctica's ice shelves have shrunk since 1997, study finds

Melting Glaciers Have Shifted the Earth's Axis - GlacierHub

What would happen if the Earth's axis was not tilted relative to the ecliptic? - Quora

Earth Axis Tilt Change Caused by Melting Ice

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 due to climate change caused Earth's axis to shift since mid-90s-Tech News , Firstpost

Glacial melting from climate change shifted Earth's poles

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

What would it take for the Earth's axis of rotation to change? - Quora

Melting Glaciers Have Shifted the Earth's Axis - GlacierHub

Melting Polar Ice Caps Distributed Enough Water to Cause a Shift in Earth's Crust Horizontally, Study Reveals

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