(Why I think the Declaration of Independence reference to "Supreme Judge of the world" isn't a deistic reference.)
It's not an obscure reference to God, but a reference to other nations.
Within the Declaration itself:
When in the Course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume among the powers of the earth, the separate and equal station to which the Laws of Nature and of Nature's God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.
... To prove [the tyranny of the English Crown], let Facts be submitted to a candid world.
We, therefore, the Representatives of the united States of America, in General Congress, Assembled, appealing to the Supreme Judge of the world for the rectitude of our intentions, do, in the Name, and by Authority of the good People of these Colonies, solemnly publish and declare, That these United Colonies are, and of Right ought to be Free and Independent States ...
This is supported by Paine's Common Sense (Of The Present Ability of America, with Some Miscellaneous Reflexions):
TO CONCLUDE, however strange it may appear to some, or however unwilling they may be to think so, matters not, but many strong and striking reasons may be given, to shew, that nothing can settle our affairs so expeditiously as an open and determined declaration for independence. Some of which are,
FIRST -- It is the custom of nations, when any two are at war, for some other powers, not engaged in the quarrel, to step in as mediators, and bring about the preliminaries of a peace: but while America calls herself the Subject of Great Britian, no power, however well disposed she may be, can offer her mediation. ...
SECOND -- It is unreasonable to suppose, that France or Spain will give us any kind of assistance, if we mean only, to make use of that assistance for the purpose of repairing the breach, and strenthening the connection between Britain and America; because those powers would be sufferers by the consequences.
THIRDLY -- While we profess ourselved the subjects of Britain, we must, in the eye of foreign nations, be considered as rebels. ...
FOURTHLY -- Were a manifesto to be published, and despatched to foreign courts, setting forth the miseries we have endured, and the peacable methods we have ineffectually used for redress; declaring, at the same time, that not being able, any longer, to live happily or safely under the cruel disposition of the British court, we had been driven to the necessity of breaking off all connections with her; at the same time, assuring all such courts of our peacable disposition towards them, and of our desire of entering into trade with them: Such a memorial would produce more good effects to this Continent, than if a ship were freighted with petitions ot Britian.
Under our present denomination of British subjects, we can neither be received nor heard abroad: The custom of all courts is against us, and will be so, until, by an independance, we tak rank with other nations.
Clearly, the concern is to convince other nations to accept the united States of America as an independent nation, and not as a part of Great Britain.
"Supreme Judge of the world" isn't a reference to God, but to other nations.