According to your references in the images attached, aragonite and calcite can well be distinguished. However, the relative intensities of the strongest peaks of calcite and aragonite is not only dependent on the amount but also properties such as crystallinity, grain and crystallite size etc. The proper way to solve this would be a Rietveld refinement. Or - if available - for qualitative comparison, you could use pure calcite and pure aragonite samples measured on your device. If the maximum intensities of the main peaks of aragonite and calcite are similar, you could directly estimate their amounts from your diffraction pattern - but only roughly.

According to your references in the images attached, aragonite and calcite can well be distinguished. However, the relative intensities of the strongest peaks of calcite and aragonite is not only dependent on the amount but also properties such as crystallinity, grain and crystallite size etc. The proper way to solve this would be a Rietveld refinement. Or - if available - for qualitative comparison, you could use pure calcite and pure aragonite samples measured on your device. If the maximum intensities of the main peaks of aragonite and calcite are similar, you could directly estimate their amounts from your diffraction pattern - but only roughly.

An important prerequisite for the quantitative X-Ray analysis is a texture-free powder preparation. If you put the calcite powder on the preparation pad, then the fragments are oriented preferentially along the surface because of the perfect cleavage of calcite. This texture effect influenced or distorted the quantitative evaluation. The addition of cork dust can reduce the texture effect.

Have you tried the selective staining method with Alizarin Red S for the determination of the calcite and aragonite ratio on the surface of your rock specimen?

https://www.researchgate.net/publication/262389656_Assessment_of_early-age_autogenous_shrinkage_strains_in_concrete_using_bentonite_clay_as_internal_curing_technique

Article Assessment of early-age autogenous shrinkage strains in conc...

Quantitative analysis of different crystallographic forms of CaCO3 like calcite and aragonite can be done by XRD and Rietveld refinement method. I got reasonably good results for my samples. Chemical analysis of samples will be useful to get an idea of Ca wt%. All the best

You can use Rietveld refinement method.

I agree with most people : you should use Rietveld refinement method (software : Topas by Bruker AXS, or FullProf or others) : it can quantify accurately the CaCO3 phases and this softwares have paramters to enter/modify in order to help getting rid of preferential orientation (like Guenter Grundmann this may be a major issue if the preparation of the sample is not good enough, but it can be solved if the orientation is not too strong by the maths in the rietveld refinement). I totally disagree with the one saying that XRD is not the good method to distinguish those two structures : XRD is the perfect method for that and she explains it herself in the beginning of her message... But I think you already know that since you used the method and everyone using this method know at least that...

Since you are only interested in an roughly estimation you could use the Reference Intensity Ratio (RIR) method. The I/Ic values you need are tabulated values, e.g. in the ICDD database. But keep in mind, that the error of this method is in the region of, at least, +-10%. If you had to exercise this task more frequently it will be a good idea to determine your own I/Ic values.

Also look at this related discussion:

https://www.researchgate.net/post/What_is_XRD_Rietveld_phase_fraction_detection_limit

The Rietveld method has all the answers you need for a rough estimate of the ratio between calcite and aragonite, as you declare you are seeking . You can also refine it to a better precision by adjusting the various parameters involving crystallite size, preferred orientation etc..

Alternately you could make yourself an experimental scale by preparing known mixtures of the two (plus the two end-members) and use the ratio between two (or more) representative peaks. This should provide you with a linear function onto which you can plot your unknown mixture and read off its composition directly.

If there is no vaterite observed in XRD analysis, you can use equation given by below for determination of aragonite content in the sample, 

y = 3.9Sa / (Sc + 3.9Sa)

where y is the calculated fraction of aragonite; Sc and Sa are the integrated intensities of X-ray diffraction spectra characteristic of calcite (d(104) = 3.3035A° , 2h = 29.404j) and aragonite (d(111) = 3.396A° , 2h = 26.213j), respectively.

i added article in attachment.

it maybe useful for you.

(sorry for late answer)

regards.

Felix is absolutely right, do quantitative Rietveld with profile refinement and you have immediately the answer you want.