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

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

Claim: Sea level rise from ice sheets track worst-case climate change scenario • Watts Up With That?

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

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

Climate change has knocked Earth off its axis, study shows

Warping of Planet's Crust: Melting of Polar Ice Shifting Earth Itself, Not Just Sea Levels

Satellite Reveals End of “Unending†N. California Drought - Roy Spencer, PhD.

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

Climate change has destabilized the Earth's poles, putting the planet in peril - The Washington Post

If all of Earth's ice melts and flows into the ocean, what would happen to the planet's rotation? – Climate Change: Vital Signs of the Planet

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

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

Climate Change Has Shifted the Axis of the Earth