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

The Earth has tilted on its axis differently over the last few decades due to melting ice caps

Melting of polar ice-caps is shifting the Earth from within, triggering movement - India Today

Scientists Discover How the Tilting and Wobbling of the Earth Influenced the Fate of Ancient Ice Sheets

Here's What Earth Would Look Like If All The Ice Melted

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

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

Polar Ice Caps Melting, Causes & Impacts - Video & Lesson Transcript

Mercury (planet) - Wikipedia

How climate change moved Earth's axis - Science

Earth's axis can affect how big Antarctica's ice sheets are - Scimex