Quantifying data completeness in healthcare In healthcare data, there are often situations when we don't know how complete or incomplete is the dataset for a given patient population. 
Let's say for example, there are 10,000 patients in the database of a hospital (Hospital A). In the community, there maybe 4 other hospitals (B, C, D and E) and tens of physician practices (outpatient clinics). Now, for a given patient, Hospital A may receive information only from Hospital B and C and only half the clinics. Since the patient may chose to go to any healthcare provider in the community, it means that Hospital A may not have complete picture of the patient's health. For example, if the patient went to Hospital D and E for last 3 emergency room visits, that information will not be available to Hospital A. Similarly, if the patient went to a clinic for regular health checkup and that practice did not send data to Hospital A, any lab work done at the clinic or vitals taken at that clinic will not be available to Hospital A.
What this means for Hospital A is not that the entire data for a variable called ER visits for a given patient is missing but that variable is underreported (instead of 6 visits, it will have only 3). Similarly, patients blood glucose level may have been taken during last visit 1 month ago but the latest blood glucose level in Hospital A's database might be from 6 months ago. 
Given this scenario, are they statistical methods that can be employed to quantify data completeness both at macro level and at patient level?
 A: The question does make some structural assumptions that we might want to unpack for readers not used to healthcare data. The poster is taking the perspective of a hospital system and, presumably, affiliated physicians or clinics. Any particular patient might get all their care at one system, or they might get care from several systems. If I am hospital A, I will have Mrs. Chan's record of interactions at my hospital and affiliates, e.g. her emergency visits, her lab test results, etc. However, I have no assurance this information is complete.
A similar problem occurs in the United States because we have many insurance providers. Even among older adults covered by Medicare, about 75% of them are in the government-run fee-for-service system, and a researcher could get access to their (de-identified) healthcare claims from the government, but the remainder are in semi-standardized private insurance plans (run under some government regulations), and we can't typically see their claims. Or, Veterans Affairs patients can and do seek care outside the VA system. Or, if someone is uninsured for part of the year, no payer database (i.e. insurance claims data) will know what happened to them while uninsured. I am focusing on the US because I live there.
The original post asked:

Given this scenario, are they statistical methods that can be employed to quantify data completeness both at macro level and at patient level?

In describing missingness, researchers have typically focused on qualitatively describing the type of missingness, e.g. missing completely at random, missing at random (I prefer to say missing conditional on covariates), or not missing at random. We'd then ask what proportion of the dependent and independent variables of interest are missing. In more advanced uses, we'd probably try to specify some mechanism of missingness. We could maybe show how much the regression parameters are biased from a naive analysis (e.g. complete cases only, or last observation carried forward if it's a longitudinal study of some sort).
In the US, some states have all-payer claims databases (APCDs), which assemble healthcare claims from all payers in the state (it's harder than it sounds, FYI). As mentioned in comments, I can foresee taking data from an APCD, and using it to predict the expected number of different healthcare systems a patient would use conditional on covariates, or something like the expected amount of doctor, emergency room, or inpatient visits attributable to a hospital system given covariates.
The thing is, in my experience, health services researchers in the United States are often not based at hospitals. They're often based at insurers; for someone insured at an insurer, all their claims will be visible to the insurer, at least as long as the person is covered by that insurer. Also, the expected proportion of missing information could vary considerably depending on which hospital you are. I'm not sure how generally useful this concept is, but I'll keep my eyes open for any applications.
I criticized the other answer as being too general. Then I wrote this one, but I think it too is still on the side of being too general. Any specifics that other commenters have will be excellent. However, I think that a much more specific answer will need more of a specific use case.
A: There are a couple of ways to handle missing data for a data set. If you understand why the data is missing, your situation sounds more like a Missing Completly ar Random (MCAR) because you have no control over where the patients go to seek care. In this case you would remove these values but it doesn't sound like this is what you want to do.
Your situation sounds more likely to be, Missing Not At Random (MNAR) in which removing the missing values will produce bias in the model, which can cause trustworthiness issue in your data set in the future. Also, eliminating those observations don't necessarily give better results.
Computing the overall mean, median or mode could be a good imputation method to use. For example, calculating the mean, median or mode from similar patient observations and using those calculations for the missing data. 
