If the target language is Haskell or Ruby or another "sufficiently DSL-capable language" it will be possible to make an internal DSL that encapsulates your query. However, in that case I think you'll surprise your users when you either have non-obvious limits to how you can intermix query with in-language code, or you'll surprise them with how badly your database performs compared to Postgres. You can see a little of both in the story of RethinkDB. If you are not using an internal DSL, you'll be stuck in the same situation as SQL where you are embedding your query language in the code somehow.
Relational databases are not just storage technology with a complicated query language. They are also integration technology. SQL's surface syntax may be unfortunate, but I'm increasingly doubtful that there is going to be a serious general-purpose database competitor that manages to hit the same notes only better. The main contender from the 90s was OODBs; they managed to have both shitty performance and lacked the ecosystem that makes SQL so useful as an integration technology: your application may be the main actor, but there are usually a large number of supporting cast members like cron jobs, reports, little scripts, ad-hoc queries, backup and restore etc, and having to do all of that with an OODB is very painful.
There are now and will continue to be niches where relational databases suffer, the prominent one today being distributed data storage. But the problem you think is a problem really isn't a problem. Industrially speaking, "I don't like SQL" is a non-problem. For people who hate it, there are ORMs that make it possible to ignore it most of the time for better or worse (mostly worse, but whatever). Syntactically, the main problem with it is, as another commenter stated, different dialects that behave slightly differently. This turns into a major engineering annoyance, but one that is mostly swallowed by your library dependencies, who have by now already basically taken care of it.
The benefit of using a modern relational database (I'm talking about Postgres but it applies to SQL Server and probably others as well) is they already have hundreds or thousands of man-years of effort put into optimizing them. I really thought RethinkDB had a brilliant design and it was going to be the next generation database. But it performed worse than Postgres and that makes it basically a non-starter. This is part of why niche databases are not based on SQL: if you handle a niche well, I will tolerate your oddities, but if you want to build a new general-purpose database today, in 2018, you can't just have a better design. Your better design has to come with better performance and the ecosystem, the tool support, maintainability, etc., or it's a non-starter. Databases are one of the hardest technical markets to break into. For most companies, the data is actually the most important thing.