In trying to generalize concepts from algebra to spectra, there are several issues that come into play.
In order for a concept in stable homotopy theory to be intrinsically meaningful it generally needs to be invariant under weak equivalence - whatever the appropriate notion of "weak equivalence" is (of spectra, of commutative ring spectra, etc). There are multiple reasons for this. On one hand, it's the homotopy category rather than the category of spectra that is "algebraic" enough to support generalizations like this. On the other hand, there is the practical consideration that there are many different models for spectra (e.g. symmetric spectra, orthogonal spectra, EKMM spectra, various diagram categories); if a concept isn't meaningful from the point of view of homotopy theory, it may have entirely different meanings in different models.
In addition, a concept may have several different directions of generalization. You could generalize an algebraic concept to one that's defined in terms of homotopy groups; this is easy to define and check, but tends to be less interesting and not satisfied in some principal cases of interest. You could try to phrase things in terms of categorical properties, and express a generalization that way; in order for this to be sensible you generally have to replace all concepts by their appropriate "derived" notions (derived pullback, derived invariants under a group action, etc), which makes it difficult to work with concepts that have almost no exactness properties. You could do something ad-hoc.
For these reasons, it's not a straightforward procedure. It's often a good idea to have some examples in mind or be looking for an application, rather than just generalizing for its own sake. A handy test for how difficult it will be is to try and determine a generalization for differential graded modules and algebras first.
Here are some of the pieces that show up in the definition of a Dedekind domain.
Integral domains. I don't really know a useful generalization that doesn't involve being an integral domain on homotopy groups. This leaves out a lot of interesting examples - there is a large zoo of regularity conditions in algebraic geometry are not satisfied by a large class of ring objects in homotopy theory.
Fields of fractions. Inverting elements - and localization in general - is something that works well in homotopy theory, and tends to give the expected results.
Integral closure. This one is much more difficult, because it involves solutions of an equation, and trying to "adjoin" elements in the fraction field. As David White mentioned, the concept of being a "subobject" is one that doesn't translate well, and so there's not a straightforward way to take an element in the homotopy of the fraction field and adjoin it to the base ring. In general it is very difficult to construct commutative ring objects with prescribed properties.
Rings of integers. See above. If you figure a out a useful notion for this, I'd love to hear from you.
Ideals. Again, there isn't an intrinsic meaning to "subobject" or "quotient". There are generalizations of the concept of an "ideal", but all the ones that I'm aware of boil down to an ideal being, by definition, something that gives you a map out to another ring. More problematically, because taking the "quotient by an ideal" usually involves a mapping cone/cofiber, being an ideal isn't a property of a map $I \to R$ of modules - it is all the extra data that allows you to construct a ring structure on $R/I$. In addition, for $R$ commutative, ideals as an associative ring and ideals as a commutative ring become separate concepts.
Principal ideals. A principal ideal, ideally, would be generated by an element in homotopy that you want to take the quotient by. Simply put, given an element in homotopy you may not be able to construct such an ideal even in cases that look amenable. If you can construct an ideal so that there is a quotient associative algebra, there are likely to be many different choices of quotient algebra structures. Being able to construct a quotient commutative algebra is an entirely different, much harder problem that often doesn't have a solution, and when we hope or expect it to have a solution we often can't prove it. There are decades-old conjectures about some of these.
Dimension. To define dimension you usually need prime ideals. There are useful definitions of dimension that use thick subcategories of the homotopy category of perfect complexes - see the work of Paul Balmer in particular. However, it is much harder to translate "deep" results about dimension into homotopy theory. More seriously, a heavy ratio of the interesting examples we know don't satisfy anything like a Noetherian property.
Having said all of this, the subject is in flux and we understand more as time goes on.
The Picard group exists very generally, for some notion of "exists". A general definition was given in a paper by Hopkins, Mahowald, and Sadofsky. For a strictly commutative ring spectrum $R$, an invertible module $M$ is one such that there exists an object $N$ such that
$M \wedge_R N \simeq R$. The category of such $M$ is the Picard groupoid; if the homotopy category is essentially small then there is an associated Picard group. This has applications in a number of areas including computational applications. You can also interpret $RO(G)$-graded homotopy groups in equivariant homotopy theory in terms of some part of the Picard group - but $RO(G)$-graded homotopy predates this work by a very significant margin.