Tarski himself thought that his semantical method could not be applied to a natural language, for the two reasons that Davidson highlights on p. 104.  The second reason (that formalizing a natural language might not "preserve its naturalness") is not compelling, but the first reason, the Liar Paradox as emphasized by Andrea A., is serious:  So long as the Liar is unattended to, a Davidsonian truth definition for English will itself contain a contradiction.
     Here's what Davidson means about "universality":  Tarski had kept the Liar out of formalized languages by stipulating that no formal language is allowed to contain its own truth predicate.  If you have a formal language L1 and you want to call one of L1's sentences true or false, you can't do so in L1 itself; L1 is not allowed to contain any term that means "true in L1."  Rather, you must speak in a metalanguage, say L2, which contains quote-names of L1 sentences and also the term "true-in-L1."  If you want to predicate truth or falsity of an L2 sentence, you can do so only in a higher-level language L3, and so on.  Thus, a hierarchy of metalanguages for formal systems.  But Tarski insisted that natural languages are not stratified in this way.  "True" and "false" are just more words of English, and apply within English to English.  English is just one language, and all English words are English, so the Liar remains a paradox.
    Davidson makes two responses to the problem.  (1) Most of the semantical questions and issues arising in the analysis of a natural language (definite descriptions, belief sentences, comparative constructions, adverbs, funny modal operators,...) do not involve truth, and illuminating solutions to such problems can be found irrespective of the Liar.  (2) Tarskian universality is suspect, precisely because it is implicated in the Liar Paradox.
    I mentioned a third response in class: that the Liar Paradox is a paradox in its own right.  As such, it's everybody's problem, not a contradiction specially generated by Davidson's Tarskian sort of theory of meaning for English.  Possible rejoinder: I can do philosophy in other areas (ethics, philosophy of mind, aesthetics, philosophy of sport) without ever coming near the Liar Paradox.  But Davidson's theory by its nature, precisely because truth is its centerpiece and it consists of trying to do just what Tarski warned against, runs smack into the paradox; it runs right up to the Liar and yells "Bring `em on."  If you do that, you do need to have some good line on the Liar.
    I've never been sure about that rejoinder.  What do you think?
    In my Davidsonian book, Logical Form in Natural Language (1984), I contented myself with a combination of response (1) above and an incomplete gesture toward my own later solution to the paradox.  I have since produced that solution (“A Truth Predicate in the Object Language,” MS).  It is a hierarchical solution with many fine qualities, and it avoids the standard objections to hierarchical solutions.  But I haven't published it, because my colleague Keith Simmons (who is one of the world's leading experrts on the semantical paradioxes) found a nasty flaw in it that I have not yet been able to repair.
    Here now is a fourth, nonDavidsonian response to the Liar objection to Davidson:  English is indeed "universal" and contains its own truth predicate.  So English is inconsistent, and so is any Tarskian-Davidsonian theory of meaning for English.  But inconsistency is not as bad we might have thought.  The great fear has always been that, since a contradiction entails anything (A & -A /- B is provable in classical logic), every contradictory theory is the same, equivalent to the set of all truths and all falsehoods.  If you accept a contradiction, you in effect accept every proposition there is, the false and the contradictory as well as the true, and rational discourse collapses.  But that is no longer true.  Logicians in (as you might guess) Australia have developed a nonclassical logic called paraconsistent logic, which tolerates contradictions so long as the contradictions are in a certain way walled off where they can do no harm.  The point of paraconsistent logic is to block the "contradiction explosion" inference from A & -A to any irrelevant proposition B.  So, even if your theory does contain a contradiction, the contradiction is in a maximum-security prison and not running amok in the streets; rational discourse can continue.
    Davidson himself would have hated the very thought of paraconsistent logic; he was very conservative about classical logic and didn't even like elementary modal logic.  But paraconsistent logic has other, less controversial uses and is, I think, here to stay.
    To be fair, it's not quite that easy.  If we ally ourselves to this fourth, paraconsistency response, we are still admitting that the actual world contains a true contradiction (the Liar sentence itself), which is still a shocker even if the contradiction does no widespread harm.  A few philosophers, most notably Graham Priest at the University of Melbourne, are quite happy with the idea, but Keith Simmons and our own graduate student Greg Littman have argued subtly against it.  References upon request.