How to validate Shopify theme settings?

Unfortunately, no: Shopify does not allow custom validation for theme settings through the admin tools

If you want to enforce constraints on any of the inputs in the theme settings, you are restricted to the controls on the data types available:

• Range: Good for number ranges that have a min/max, but resolution is limited to the step that you specify. Shopify also has a max of ~100 stepping-points between min and max, so if the range you have to cover is large enough this choice won't cover you

• Select: Best for text options, and quickly becomes cumbersome if you have more than 6-8 options.

• Radio: Also best for text options, and only where there are only a small number of choices.

If these limited use cases can't cover your validation logic, then unfortunately there isn't anything you can do to specify your own validation formula that will be enforced during theme setting updates.

There is a (slightly hacky) way to show an error only in the theme customizer, however

Using this trick shared in the Shopify Community, you can check the contents of content_for_header to test for one of the scripts injected when viewing the theme through the customizer preview:

{% if content_for_header contains "Shopify.designMode" %}
  Customizer
{% else %}
  Live!
{% endif %}

This would let you write a snippet that could test any settings with complex validation logic and display a big ol' error bar on the theme customizer page if the settings don't check out while not alarming any shoppers on the actual site.

Note that this will rely on the injected code that you test for to be present - if Shopify updates their code injection in the future this trick would stop working/would need to be updated to test for whatever they update the customizer code injection to.