Why can't we say a circle or some part of a circle is a straight line?

There are a few simple reasons why straight lines and circles are treated as fundamentally different things. A straight line is the shortest path between two points (i.e. a geodesic) in flat, Euclidean spacetime. In addition, a straight line is uniquely determined by any two points through which it passes: for any two points in flat, Euclidean space, there is only one straight line that passes through both of them (if we assume that the straight line is of infinite length so that we are not treating straight lines of different, finite lengths as distinct). A circle is not the shortest distance between two points in flat, Euclidean space, However, the shortest distance between any two points on the surface of a sphere is a great circle (a circle of radius equal to that of the sphere, so that it cuts the surface of the sphere into two equal pieces), and this great circle will be unique (just like the straight line case in a non-curved space). But on a flat 2D surface, there are infinitely many circles, of varying radii, passing through any two given points, unlike the unique straight line connecting those same two points. Of course, it is true that a finite segment of a sphere can be made to approximate a straight line as closely as desired by letting the radius of the circle tend to infinity, so that its curvature tends to zero. A segment of the circle containing a given point would then become almost indistinguishable, for all practical purposes, from the tangent to the circle at that point (i.e. the straight line just grazing the circle at that one point), and we may then choose to replace the one with the other to perform calculations. But remember that this is only an approximation; we can never say that the segment and the tangent are identical. And remember that, however large we make the radius of a circle, it must nevertheless remain a closed curve, so that one could travel in the same direction all the way around it from any point and return to that same point (even if only after an infinite amount of time). A straight line, however, is an open curve, and traveling along it in one direction from any given point would never return you to the same point; you would just keep on going in the same direction forever (for an infinite line). So circles and straight lines have a fundamentally different topology. Also, any circle drawn on a 2D plane will divide the plane into two distinct pieces, one piece being interior to the circle, and the rest being exterior. The same is not true of a straight line (though it could be argued that an infinite straight line will still divide the plane into two pieces (see comments below)).. For these reasons alone it is necessary to treat straight lines and circles as distinct, unless we are happy to use straight-line approximations for segments of circles with very large radii, and to ignore questions of topology.

One key difference between circles and lines is that lines have zero http://en.wikipedia.org/wiki/Curvature, whereas circles have constant nonzero curvature.  There are contexts where it makes sense to talk about circles and lines together as curves with constant curvature, but you're collapsing a lot of generally important distinctions when you do so.

If the circle is infinitely big, I think you can, but I am a nerd. In mathematics there is field called topology. And in that field a circle is homeomorphic to a square. If you think about a circle that is infinitely big and a square that is infinitely big you can overlay both of them. So any section you examine will be found to be a straight line.

Ifyou are working on a spherical "plane", you might consider that a straight line that is a part of a great circle of the sphere is simultaneously a circle when observed from outside the spherical frame of reference, while it is entirely a straight line related to the spherical frame of reference.

It is not true on a plane.  One can show by calculus of variations that the shortest distance between two points for a curve lying on a plane is the straight line segment connecting the points.