Actually, the plaintiff’s lawyer could. He may not choose to. Also, the court may not allow him to. Although he could. And it could.
Federal Rules of Evidence Rule 611
Edit to include more detail, as suggested in the comment:
For instance, in the absence of any evidence indicating that alcohol was involved, I would imagine that a plaintiff's lawyer in an accident case could not cross-examine a defendant with leading questions suggesting that he had been drunk at the time of the crash.
Actually, the plaintiff's lawyer could. According to FRE 611(c), leading questions should be allowed "(1) on cross examination; and (2) when a party calls a hostile witness, an adverse party, or a witness identified with an adverse party."
However, he may choose not to if it does not advance his case.
Also, the court may not allow him to... but, it could. According to FRE 611(a), "The court should exercise reasonable control over the mode and order of examining witnesses and presenting evidence...
If it determines a line of questioning is way off base, FRE 611(a)(3), providing a reason why a court would exercise such control over the examining of witnesses, lists "[to] protect witnesses from harassment or undue embarrassment." Arguably, if you were implicated as being a drunk driver when, in fact, you were not, that could constitute harassment or undue embarrassment.
Expanding on both what the lawyer may ask as well as what control the court may exercise over the questioning, FRE 611(b) says that "[c]ross-examination should not go beyond the subject matter of the direct examination and matters affecting witness's credibility. The court may allow inquiry into additional matters as if no direct examination."
So, the plaintiff's lawyer could cross-examine a defendant with leading questions suggesting that he had been drunk at the time of the crash. Why not? @bdb484, in the answer your provided yourself, it appears you are putting too much weight onto the idea of the lawyer having a reasonable basis to believe something. This is a low bar. If you were at a bar, left the bar, and got in a crash, there would almost certainly be a reasonable basis to inquire as to whether you were drunk.
The rule you cite includes,
Innuendoes and insinuations of inadmissible or nonexistent matters are improper.
Note that it does not bar those insinuations based on evidence not admitted at the trial. Rather, it bars those insinuations based on evidence that can not be admitted at the trial.