I think the acts of god on earth might be measured sciencifically, but they would be only of a part of what most consider god/the divine. The whole idea of an omnipotent, omniscient, omnibenevolent, and eternal being means it can't be measured, so the bits and pieces that interact with the natural world be what is left.
The whole idea of an omnipotent, omniscient, and omnibenevolent being is a logical flaw, you cannot have all three. You really can't even have two of the three.
It comes down to this, if it exists, it is measurable. I can even measure emotion by relative strength if not by unit, I love my son more than I love pancakes.
Scientific study needs a repeatable, measurable phenomenon, for instance, dropping a ball. You can do it over and over, you can measure speed, height, rate of acceleration etc.
To measure god/the divine, you would need a repeatable, measurable miracle. I can think of several reported miracles that are repeated, most of which only happen once in a lifetime: the incarnation of a new lama, a new prophet in the Mormon church or the seating of a new pope, etc.
There has never been a proven and documented miracle.
The problem is that, if any given religious doctrine was in fact, correct, miracles would be repeatable and measurable.
Per the bible, if you are a true believer in Christ, you can safely drink poison. Based on this, either no one on earth is a true believer, or the bible is in error. Based on biblical doctrine and the teachings of the church (namely, god is infallible) I could at this point claim I have wholly disproven Christianity. Or I've wholly disproven reality, one or the other.
This is the difficulty I have with the whole concept of religion. All religions are internally inconsistent with each other, and with a little effort (very little, in some cases) can be disproven via their own dogma. Which brings me back to my original point, what is the definition of 'god'? What is 'god'?
If I take an anthropological view of 'god', one that includes folks like Zeus and Ba'al and Neptune, then turn to the bible, there are several gods, and not just Jesus and Jehovah. You could relegate folks like Goliath and Moses as demigods, but they have traits that would demonstrate them to be 'divine'.
Which brings us to the major reason I'm not religious. I question things and don't accept answers that do not work with reality. I don't take anything 'on faith', as it were. If something flies in the face of logic, rationality, and truth, I question it, which is incidentally also the reason why I reject membership in any political party.
One miracle that is said to happen virtually daily, all over the world, is the Roman Catholic Transubstaniation, the literal transformation of the bread and wine into the body and blood of Christ. However, only the substance is altered, not the physical breadness of the bread. I don't quite understand the arguments about the way it is supposed to work, but it also gets around the cannibalism issue. However, I don't know how you would set up an instrument to measure the philosophical difference between blessed and unblessed host, since it is not an physical change. It would also test only an aspect of the divine, not all of the godhead. If you could disprove transubstaniation, it would not disprove god, only that aspect of god.
It doesn't happen. The substance is bread, and it remains bread. The substance is wine, it remains wine. The SYMBOLISM is the important part, and the part that matters.
Prayer is also one that receives a great deal of attention. Some of the human aspects have been documented. There are changes in the brain that happen with prayer, especially repetitive prayer or chanting. I don't think this is the god/the divine; I think it is an aspect of humanity. There would be no way to measure if prayer were answered. What would be the variables, what would be measured, would the person praying make a difference, would the deity being prayed to, would the language of the prayer make a difference, would the number of repetitions, how would you determine the prayer had been answered.....lots of questions, way too many variables.
The same changes in the brain happen when cheering at a pep rally.