Stabilization of a generic pointed model category Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}_*)$-category, where $\mathrm{sSet}_*$ is the Quillen model category of pointed simplicial sets.
With this structure one can define suspension and loop functors on $\mathrm{Ho}(\mathcal C)$. The suspension $\Sigma$ is the (derived) smash product with the simplicial $\mathbf S^1$ and the loop functor is its right adjoint. By definition, $\mathcal C$ is called stable iff $\Sigma$ is an autoequivalence of $\mathrm{Ho}(\mathcal C)$.
In case $\mathcal C$ is not necessarily stable, I'm looking for canonical ways to stabilize $\mathcal C$. In case $\mathcal C$ fulfills extra axioms, e.g. it is left proper and combinatorial, one can find by [1] a simplicial model structure on the simplicial objects $\mathrm s \mathcal C$ of $\mathcal C$ which is Quillen equivalent to $\mathcal C$ itself. So for left proper and combinatorial model categories $\mathcal C$, one may assume that they are already simplicial so that the suspension is already defined on the level of the model category. Then one can use [2] or [3] to consider (symmetric) spectrum objects in $\mathcal C$ with respect to the suspension and gets a canonical stable model category, where the model structure is the stable one.
Now my question is as follows: Is there a canonical way to stabilize a model category without going through simplicial enrichments (and which may work with any closed model category)? Bonus questions are: Are there better or more canonical references than the two I gave? Does it make sense just to stabilize the homotopy category $\mathrm{Ho}(\mathcal C)$, e.g. just to consider spectra on the homotopy level?
[1] Dugger: Replacing Model Categories with Simplicial Ones
[2] Hovey: Spectra and symmetric spectra in general model categories
[3] Schwede: Spectra in model categories and applications to the algebraic cotangent complex
 A: This question is near and dear to my heart. Since it is asked within the context of model categories I will try to answer in that context. I agree with Dmitri that you should not try to stabilize the homotopy category. Your main question seems to be about whether or not $C$ needs to be simplicial. The answer is no. Hovey's machine does not require a simplicial model category, but requires you to have the functor you want to stabilize. Via the machinery of framings (see chapter 5 of Hovey's book) you can write down that functor even when $C$ is only cofibrantly generated. This is the approach taken in chapter 6 of Hovey's book. Next, let's discuss the hypotheses that $C$ be left proper and cellular or combinatorial.
As far as I am aware all stabilization results involve Bousfield localization. However, the version of Bousfield localization which Schwede uses (improved upon in Bousfield's paper on Telescopic localization) does not require combinatoriality or cellularity. Those hypotheses are present in order to actually build the localization functor, which works via some transfinite process you need to know will eventually end. But if you already have the localization functor then you don't need those hypotheses. So I think what you propose can be done, but you will want to be careful to determine which model of spectra you want. As Hovey points out, Schwede's machine and Hovey's machine build different model categories of spectra, and only Hovey's appears to have the property you want. So basically you would want to take Hovey's proof and make it go with a localization functor built "by hand" so as to avoid the need for $C$ to be combinatorial or cellular. Note that without left properness you can still say something, and I have a preprint on that if you would like to email me. It's not yet ready to write about publicly here. I advise working out an example, e.g. take $C$ to be the stable module category and figure out what the Ho(sSet) action does. Use a framing to write down the suspension endofunctor and try to build the localization following Hovey and Schwede. If it works then you have a nice roadmap for what to do in general, and the material in chapters 5-6 of Hovey's book will help fill in the details.
There are also other ways to stabilize, e.g. Biedermann's paper L Stable Functors, which basically defines the stabilization you want, tries to build it, and has a comparison to Hovey's machine. The relevant results are in section 5, $M$ is what you called $C$, and $V$ can be simplicial sets or chain complexes or something else. This uses the version of Bousfield localization I mentioned above, but it (and also Proposition 2.2 of Motivic Functors by Dundas, Rondigs, Ostvaer) technically require locally presentable in order to embed into a presheaf category. Perhaps this can be avoided; I advise asking a category theorist.
Since Dmitri mentioned Higher Algebra, let me just say that I don't understand the proof of 1.4.2.24. It first does the case where $C$ is presentable, which we already knew how to do thanks to the references in the question, and then it says without loss of generality we can assume $C$ is small. I don't follow this step. Perhaps it's one of these arguments where you enlarge the Grothendieck universe but in this context it seems to be getting something for free. In any event, this step is probably not replicable in the context of model categories because once you bump up to a larger Grothendieck universe I don't know what happens. If the model category truly becomes small then it's uninteresting (e.g. just a poset), but I think what really happens is more in line with the discussion at Is the $\infty$-category of presentable $\infty$-categories presentable?, i.e. in this new universe $C$ won't have all limits and colimits, only those of some bounded size. In any event, I don't think you can reduce questions about model categories to dealing with small ones.
