How to justify that a significance test cannot support the null hypothesis? It is always emphasized that a significance test cannot prove a null hypothesis, only disprove it. Do you know any authoritative source (like a text book in statistics) that explains it and could be cited in a scientific paper?
The problem is that in validation studies (e.g. when validating that a new device is accurate), significance tests have been commonly used for PROVING null hypotheses. The logic is to prove that X=Y (measurements and real values do not differ) by fitting linear regression, calculating the coefficient of determination (R^2) and performing t-tests with the null hypotheses that the slope is 1 and the intercept is 0. Then if both tests give insignificant results (with p>0.05) and R^2 is sufficiently large (e.g. >=0.90 or >=0.80), it is concluded that the measurements were accurate. The data size is often very small, only 10-15 data points. 
 A: Doing this detour of a linear regression looks unnecessary to me if you can get paired data. You could also just subtract the paired measures of both instruments and have a single column of data to do a t-test on.
If you want to understand the reasons for not being able to prove the null hypothesis, read up on power analysis. It can broadly be framed as a problem of conflict of interest. As a researcher, you have a research hypothesis in mind that you want to prove. This is $H_a$. In order to arrive there, you need to produce evidence that rejects $H_0$. (Per set theory, $H_a$ and $H_0$ combined need to cover all possible outcomes.) This evidence needs to be strong enough for your p-value to express that the data would have been unlikely to occur under the assumption that $H_0$ is true.
If you want to prove $H_0$, you will be rewarded for sloppy work. The less evidence you collect, the more likely you are not able to reject $H_0$, which was your goal in the first place. In the worst case you actively ignore existing evidence against $H_0$. In the extreme, a researcher that never collects any data at all, will be able to prove all his precious $H_0$ research hypotheses and get published without doing any actual work.
Due to the mathematical nature of such tests, you can also not simply switch $H_0$ with $H_a$. For your purpose of showing a quantity does not differ from a given value, equivalence testing has been developed. It has the disadvantage that you need to define a parameter $\delta$ which represents the maximum acceptable distance to the value of interest the measures can have to still be considered equivalent. With two one sided tests (and a Bonferroni correction), you can show that your sample mean is neither smaller than the value of interest - $\delta$ nor bigger than it + $\delta$, thus equivalent.
That $\delta$ needs to be defined before looking at the data. Otherwise you have a similar conflict of interest again, the statistician would just select the smallest available $\delta$ for which he can prove equivalence. Women are as heavy as men if you take a $\delta$ of +-15kg around the weight of the average man... In this case, the scale is intuitive and the scam obvious. In most research scenarios, you could fool a reader by manipulating $\delta$. Equivalence testing is acceptable if you have principled reasons to justify $\delta$, otherwise it does violence to the epistemic principle of hypothesis testing.
A: In the specific context of validating a new device, regression can make a lot of sense. As a reviewer I would want to see analysis of a calibration curve comparing the results from the new device against known values (or values determined by a highly accurate and precise method). This would be important for documenting the limits over which the output from the new device bears a linear relation to true values and for demonstrating how the magnitudes of measurement errors are related to those values.
If the comparison can only be done between two measurement methods that both have appreciable errors, analysis of what's known in the medical literature as a Bland-Altman plot (Tukey mean-difference plot) could be more appropriate. In this context of comparing devices, restricting analysis to t-tests of paired differences between measurements would throw out very important information about how differences might change over the range of measurement.
In any case, however, you never "prove the null hypothesis" even if that's the way some uninformed people might casually put it. You simply document that the output from the new device is unlikely to differ enough from the true value (or from the results provided by the older device) to make an important difference. Calibration via the method noted in the question effectively makes some hidden assumptions about how big a discrepancy makes an important difference. Proper analysis would provide confidence intervals for the differences between the outputs of the devices. For more background on this type of problem, you might want to look into non-inferiority testing.
A: Here is one source (from medical science) that gives a counter-example that the linear regression method doesn't work:
Abhaya Indrayan: Clinical Agreement in Quantitative Measurements
Limits of Disagreement and the Intraclass Correlation, Chapter 2 in Methods of Clinical Epidemiology, Springer 2013.
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwjOofXPvqDWAhVoCZoKHWvIDMMQFggvMAA&url=http%3A%2F%2Fwww.springer.com%2Fcda%2Fcontent%2Fdocument%2Fcda_downloaddocument%2F9783642371301-c2.pdf%3FSGWID%3D0-0-45-1402889-p174984284&usg=AFQjCNHtxad27T-bNsQhZDC5jbiEkoTjJQ
In the example, one can fit a perfect linear regression line y=x (with correlation r=0.945) but still there is no agreement in any of the subjects. The errors were considered substantial in this context.
