The monopolized market in this case would be the market for default search engines on mobile.
If the only two players (Google and Apple) both use Google Search by default, then Google has effectively captured 100% of the mobile ad market. (And since Google is paying Apple et al for that default state, it is indeed a market.)
The court could require device makers (including Google itself) to prompt users to select their search engine provider, or potentially ban Google from buying search engine defaults from other companies (Apple, Mozilla, etc.) as an anticompetitive practice.
That's an interesting scenario. Let's say Google is banned from buying search engine defaults. Then Mozilla would no longer be able to sell the Firefox default to Google.
Sure, they could sell it to someone else, but without the biggest player bidding up the price, it'd probably sell for a fraction of what it does today -- which could be devastating to Mozilla, given that almost all of their funding comes from the search deal. What would that do to competition in the browser space?
Competition in the browser space is over and Chromium won. If this had been regulated earlier by preventing Google from preinstalling Chrome on all Android devices, we wouldn't be in this situation where Firefox is on life support, but here we are.
Using Firefox as a reason not to break up some of Google's hold on search seems extremely short-sighted. Besides, Edge using Chrome changed the whole browser market dynamic as more and more people are using it. Once Microsoft reaches a somewhat decent percentage of install we will be back at a two (three?) player situation.
Apple could actually make more money. Say Google is split into two renamed search engines: elgooGA and elgooGB, with evenly divided assets. They would immediately be two search titans competing for the important Apple contract and would try to outbid one another. Perhaps elgooGA wins Apple, but then elgooGB wins Firefox. Every alternative platform to Apple would stand to get more money. And the users would then become more accustomed to having different default search providers and usage might be split. That is a win for DuckDuckGo and all other competitors.
If the only two players (Google and Apple) both use Google Search by default, then Google has effectively captured 100% of the mobile ad market. (And since Google is paying Apple et al for that default state, it is indeed a market.)
The court could require device makers (including Google itself) to prompt users to select their search engine provider, or potentially ban Google from buying search engine defaults from other companies (Apple, Mozilla, etc.) as an anticompetitive practice.