This should work in theory, but the accuracy would decrease with increasing diameter:
Your image shows measuring a chord of the circle at an unknown distance from the center of the circle.
If you place a gauge block (of known thickness) between the beam of the caliper and the circumference of the tube, you can measure the length of a second chord of the circle at a known radial displacement from the first chord.
The diameter of the circle should be obtainable from the lengths of the two chords and the known radial distance between them.
The illustration at the top of this page is applicable:
The result would be affected by the roundness of the tube, surface roughness on the circumference, the perpendicularity of the caliper to the axis of the tube, and the accuracy with which the thickness of the gauge block is known.
I haven't done this, worked out the formula, or determined how sensitive the result would be to the variables listed above, but it might be an adequate and inexpensive approach in some cases.