If the image is protected by copyright (as most modern images are), the uploading it as a profile picture or using it on a web site is copyright infringement unless one has permission from the copyright owner, or an exception to copyright applies. If copyright infringement occurs, the owner can sue for damages, or issue a takedown notice, or both, but does not have to do either. In the vast majority of cases, no one will do anything about such infringements if the owner (or the owner's authorized agent) does not. All the above is true in all of Europe and North America, ad in most other countries as well.
Permission can be granted directly by the copyright owner to a user, or granted by the owner releasing the image (or other wrk) under a license. Google's general developer TOS Document](https://policies.google.com/terms#toc-content) reads (in relevant part):
Some of our services include content that belongs to Google — for example, many of the visual illustrations you see in Google Maps. You may use Google’s content as allowed by these terms and any service-specific additional terms, but we retain any intellectual property rights that we have in our content. Don’t remove, obscure, or alter any of our branding, logos, or legal notices. If you want to use our branding or logos, please see the Google Brand Permissions page.
Finally, some of our services give you access to content that belongs to other people or organizations — for example, a store owner’s description of their own business, or a newspaper article displayed in Google News. You may not use this content without that person or organization’s permission, or as otherwise allowed by law. The views expressed in other people or organizations’ content are theirs, and don’t necessarily reflect Google’s views.
It would seem that each Google service has its own terms which indicate in what ways one may use Google content. It would further seem that content originally obtained from others and distributed by Google will have some set of permissions granted by the uploader. These must be checked for each image, I would think. There is no single one-rule-fits-all answer.
I gather that the APIs will have methods for determining which images are from Google itself, and which from others, and some method for indicating the permissions that accompany an image. A user of these APIs must learn how to extract ownership and permissions metadata for an image obtained via the API, and how to use it (or not use it) in accordance with the available permissions.
Now let us look at exceptions to copyright. These vary by country. Several European countries have an exception for "personal use" which might cover the profile pictures, but probably will not cover general use on a web site. There are exceptions for "news reporting" which might cover images illustrating recent news events. Some have exceptions for educational or instructional use. The UK (and several othe countries) have the concept of fair dealing which bundles several exceptions together. This might cover an image used in a personal profile, use of images in news reports, and limited educational use, but will probably not cover general use on a website. The US has the concept of fair use which is rather complex, but does not specifically include a "personal use" exception. See Is this copyright infringement? Is it fair use? What if I don't make any money off it? and other threads here tagged with fair-use for more on this.
It is somewhat unlikely that any of these exceptions will cover general use on a web site of images obtained via Google APIs and used beyond the permissions listed in the metadata accompanying the image. But it will very much depend on the specifics of the case.
If the image fetching is doe not by you in building a site or an app, but by your app's users who may then use them improperly, strictly speaking that is each user's own responsibility, you cannot effectively police what your users do, and have no duty to try.
It will be a good practice if you include text explaining to your users that not all images are free for general use. You might want to link to the Google TOS. If the API returns metadata indicating the permissions status of each image fetched, yo0u might well want to provide this metadata in a useful form to your users. You might even want to allow a user to filter images obtained by permission status. But whether this is practical or useful to users will depend on the purpose and design of an app.