No. They must pass a state bar exam in order to be admitted to practice in their state (or DC).
Graduation from law school / awarding of a J.D. and being admitted to the bar are two different things. So is graduation from medical school / awarding of an M.D. and being licensed as a physician by a state.
A J.D. or an M.D. is an academic degree and is awarded by a university. Bar admission or physician licensure has direct legal implications and is awarded by the state or by a state-sanctioned private organization.
Earning a professional degree in medicine is necessary but not sufficient to be licensed as a physician in US states these days -- an independent license also requires at least a year of supervised practice (internship) among other requirements (passing certain tests).
Similarly, earning a professional law degree is usually among the requirements for sitting the bar exam or exams to be admitted to a state bar, but some states don't require it, offering alternate apprenticeship-like pathways. Conversely, there are countries where a law degree and exam are not enough, and a year or more of supervised practice of law is required.
By any other branch of academia except law, a medical school program which awards an MD would not qualify for a doctorate since there is no requirement for an original contribution to our knowledge.
In the most traditional (ie Latin word root) sense, a doctorate is an award to someone who is now qualified to teach at the highest university level, and doesn't directly require original contribution to knowledge (though that is reasonably enough a requirement in most of the academic disciplines). As MDs and JDs are equipped to teach in university medical and law schools, the descriptor fits.
That said, this broad definition has lead to a number of different definitions of "doctorate" in different cultures and times. It wasn't long ago that in Germanic and Scandinavian countries, a doctorate was awarded only upon reaching what we would now consider the full professor level. This is still sort of how honorary doctorates are given out, for contribution that marks the bearer as an expert in the field, and also persists in the "higher doctorates" of some countries.
Schools in the UK, which give MBBSs as undergraduate (professional) medical degrees, do also award MDs or DMs for original clinical research projects, much the same as a PhD or DPhil, but in a medical department rather than a more traditional academic one.
I guess my point is that overall, you're not going to apply logic to a system of man-made labels that have developed over about a millennium and come out ahead. In the oldest universities, admission was permission to hang out in the relevant city for a few years and chat with the academics and read in the libraries. At the end of your few years, you submitted to examination, whether by written and oral answers to questions (for a bachelor's or master's) or by submission of a thesis and oral examination (for a doctorate). And really weird things persist.
For example, at Oxford, anyone who graduates as an undergrad with a 3-year bachelor's degree (which is now not the done thing in sciences, which are all 4-year undergraduate master's degrees), you automatically are awarded an MA the following year, upon writing a letter asking for it. This is because traditionally, children were sent to live and study at the age of 12-ish, and left at the age of 20-ish with an MA, so tradition and rules dictate that a master's level graduate is a member of Convocation (the body of alumni). Once modern bachelor's degrees became the norm, there needed to be a way to allow alumni into Convocation, so an automatic MA was decided upon. And there are even weirder permutations of that.
The point is, if you want to know what somebody's done, you're going to have to go deeper than the letters after their name. And you're never going to get something as old and entrenched as academia to follow sensical rules, especially internationally.