Washington State's governor announced indoor gatherings are prohibited. On what legal basis could they be enforced in private settings (such as people in their own homes)?
When the U.S. Constitution was written and ratified, it delegated certain powers to the federal government and reserved the remainder to the states. Among the powers reserved to the states is the "police power," which is an amorphous but exceedingly broad category of powers:
Whatever differences of opinion may exist as to the extent and boundaries of the police power, and however difficult it may be to render a satisfactory definition of it, there seems to be no doubt that it does extend to the protection of the lives, health, and property of the citizens, and to the preservation of good order and the public morals.
The police power is intentionally construed very broadly, and it generally does not distinguish between public and private conduct, nor between individual and business conduct. If conduct is a potential threat to "the lives, health, and property of the citizens," it is generally fair game for an exercise of the police power.
However, when we say that the police power is reserved "to the states," we generally mean to the legislative body of the state, so any power the governor had to exercise the police power in this way should come from a legislative enactment, unless perhaps the state constitution grants him broader emergency powers.
As I understand it, Governor Inslee is relying on RCW 43.06.220, but that law does not seem to grant him authority to ban the private, indoor gatherings you've asked about. It does allow him to prohibit people from "assembling or gathering on the public streets, parks, or other open areas of this state, either public or private," but the vast majority of indoor gatherings are not going to be in "public streets, parks, or other open areas."
So my first impression -- not having studied the issue all that closely -- would be that RCW 43.06.220 does not authorize the governor to impose such restrictions on private indoor gatherings, though I could easily imagine other provisions that would grant that authority.
And -- despite the skepticism of all the other answers -- your instinct that there may be something making this legally inappropriate is correct. The issue all the other answers seem to avoid addressing is whether the law violates the First Amendment's right to assembly. While the government is permitted to reasonable, content-neutral restrictions on the time, place, and manner of public assemblies, its authority to infringe on First Amendment rights is narrower when it comes to gatherings in private property, especially in the home.
In the end, I think the question would come down to whether the governor's order passes strict scrutiny -- i.e., whether it advances a compelling governmental interest and whether it is the least restrictive means of doing so. The order would easily survive on the first prong, as preserving public health is a compelling government interest, but the second would be a lot trickier, and would likely require quite a bit of expert testimony.
If the government could achieve similar reductions in the spread of COVID-19 by requiring people assembling in private to wear masks, use disinfectant, get tested in advance, etc., then it may be that the order would fail a First Amendment challenge. But given the nature of the emergency, I suspect that if the state could make a reasonable showing that this is the best it can do given the current state of research, a court would be a bit more deferential than it would in normal cases.
Here are the Washington state proclamations with legal force, for example this proclamation amending proclamation 20-05. The legal authority is the paragraph starying "NOW, THEREFORE, I, Jay Inslee, Governor of the state of Washington" and what follows is what is legally proclaimed, in this case the effect is to "waive and suspend portions of Title 79 RCW that require in-person meetings", pertaining to Dep't. of Natural Resources. Here is 20-25, the original stay-home proclamation. It claims Chapters 38.08, 38.52 and 43.06 RCW as authority, and
prohibit[s] all people in Washington State from leaving their homes or participating in social, spiritual and recreational gatherings of any kind regardless of the number of participants, and all non-essential businesses in Washington State from conducting business, within the limitations provided herein.
43.06.220 is the main hammer that the governor can wield. The powers granted by the legislature include:
(1) The governor after proclaiming a state of emergency and prior to terminating such, may, in the area described by the proclamation issue an order prohibiting: (a) Any person being on the public streets, or in the public parks, or at any other public place during the hours declared by the governor to be a period of curfew; (b) Any number of persons, as designated by the governor, from assembling or gathering on the public streets, parks, or other open areas of this state, either public or private; (c) The manufacture, transfer, use, possession or transportation of a molotov cocktail or any other device, instrument or object designed to explode or produce uncontained combustion; (d) The transporting, possessing or using of gasoline, kerosene, or combustible, flammable, or explosive liquids or materials in a glass or uncapped container of any kind except in connection with the normal operation of motor vehicles, normal home use or legitimate commercial use; (e) The sale, purchase or dispensing of alcoholic beverages; (f) The sale, purchase or dispensing of other commodities or goods, as he or she reasonably believes should be prohibited to help preserve and maintain life, health, property or the public peace; (g) The use of certain streets, highways or public ways by the public; and (h) Such other activities as he or she reasonably believes should be prohibited to help preserve and maintain life, health, property or the public peace.
One could argue that some order is not reasonable, and perhaps it would fail strict scrutiny, if anyone sued.
At present, though, there is no legal order. There is a guidance, a statement of best practices. There may well be a formal proclamation on this topic, which would wrap this in the legalities of other official proclamations. It boils down to RCW 43.06.220(1)(h).