TL:DR Not really possible, no
This is from the website of Allen Allen Allen & Allen, a Virginia-based legal firm. (NOTE I will not be including links within the quotes, you may view their website for more information.)
In order to prevail in a personal injury lawsuit arising from the Coronavirus, the plaintiff will need to establish: (1) that the wrongdoer had a legal duty to the plaintiff (i.e., that the wrongdoer had a responsibility not to expose the plaintiff to the Coronavirus); (2) that the wrongdoer breached their legal duty; (3) that the wrongdoer’s actions caused the plaintiff to get sick; and (4) that the plaintiff suffered damages (i.e., illness, medical expenses, lost wages or death).
First, there is Legal Duty, which they show as:
In Virginia, every person has a duty to exercise due care to avoid injuring others. The scope of the duty varies with the circumstances of each case, but as a general rule, everyone owes a duty to every person within reach of their conduct. This duty “arises from that basic and necessary regulation of civilization which forbids any person because of his own convenience, to recklessly, heedlessly or carelessly injure another.“
Before imposing liability against a defendant, the court will examine whether the defendant knew, or should have known, that his conduct placed others at risk of harm.
Virginia courts have imposed liability against a farmer who allowed his cow to wander into traffic (which caused a crash) and a manufacturer whose employees wore their work clothes home (which exposed their family members to asbestos fibers and caused them to develop asbestos-related diseases). Similarly, any person who knows that they have the Coronavirus and exposes others to it could, in theory, be held liable.
Next, we have causation, which would be the major sticking point.
Under Virginia law, plaintiffs are required to prove that the defendant caused their injury. In Coronavirus cases, if the plaintiff cannot prove by a preponderance of the evidence (that it is more likely than not) that the defendant was the source of the disease, the plaintiff will not prevail at trial.
Virginia courts allow plaintiffs to pursue claims against defendants who caused them to become sick or develop disease.
Unlike the cases described above, where the source of the exposure was known and easily determined, proving causation in Coronavirus cases is challenging because an individual who is exposed to the virus might not develop symptoms for up to two weeks, and the Coronavirus can survive on surfaces for several days. Therefore, a plaintiff who did not practice social distancing or limit his or her exposure to others could find it difficult to establish that the defendant was the source of the virus.
I have left out two example cases that they show within the above paragraphs, for brevity. Both are related to passing AIDS from an HIV-positive person to a healthy person.
Finally, they talk about did the person know they have COVID-19 (positive screening for the virus) and did they take reasonable precautions:
In litigation arising from the COVID-19, courts will consider whether the defendant knew or should have known that they had the virus, and whether they knew or should have known that they would (or could) spread it to others. Certainly, anyone who intentionally and deliberately spreads the virus to others could be held liable and might face criminal penalties, as well. In some cases, individuals who have been officially diagnosed with the virus could potentially be held liable if they do not take reasonable steps to prevent spreading it to others.
However, it is unlikely that a court would impose liability against any person who was not officially diagnosed with COVID-19. It is also unlikely that a court would impose liability against individuals who take reasonable steps to prevent spreading the virus to others.
So, in my example from my question, if I don't know that my neighbor was diagnosed, or more importantly, if they don't know, then the case would go nowhere. If they received notification of a positive result of a test AND they went out shopping and did not wear a mask, then PERHAPS this requirement would be met.
However, the real sticking point here is number 2, causation. I would have to prove that I caught the virus from that person and not from any other people I may have come into contact with.
So, while I am not a lawyer, I can understand that it would be next to impossible to prevail with this lawsuit, unless I were able to show that I came into close proximity with only one person, and that that person has the virus, and knew they had the virus, and purposefully did not use precautions to protect others.
That seems an awful lot to overcome.