img = Import["https://i.sstatic.net/gZ8oj.png"]

Export[
    "TestFile.ZIP"
    ,{
        img
    }
    ,{
        "ZIP"
        ,{
            {"image.dcm","DICOM"}
        }
    }
]

In your case, change the list {img} with all the image objects and {{"image.dcm","DICOM"}} with the file and format names.

Export "ZIP"

As explained by @Szabolcs, "GZIP" will not accept more than one file, so here I show an example with "ZIP".

Where I use a single image, you can have many.

img = Import["https://i.sstatic.net/gZ8oj.png"]

Export[
    "TestFile.ZIP"
    ,{
        img
    }
    ,{
        "ZIP"
        ,{
            {"image.dcm","DICOM"}
        }
    }
]

In your case, change the list {img} with all the image objects and {{"image.dcm","DICOM"}} with the file and format names.

CreateFile

Alternatively you could use CreateFile with a similar syntax.

img = Import["https://i.sstatic.net/gZ8oj.png"]

Export["TestFile.ZIP",{img},{"ZIP",{{"image.dcm","DICOM"}}}]

In your case, change the list {img} with all the image objects and {{"image.dcm","DICOM"}} with the file and format names.

img = Import["https://i.sstatic.net/gZ8oj.png"]

Export[
    "TestFile.ZIP"
    ,{
        img
    }
    ,{
        "ZIP"
        ,{
            {"image.dcm","DICOM"}
        }
    } 
]

In your case, change the list {img} with all the image objects and {{"image.dcm","DICOM"}} with the file and format names.