Given, say $1sec$ of measurement data, which may contain a wide range of frequencies, we fit an $n^{th}$ order polynomial curve through it. Doing this filters out certain frequencies which would have been present otherwise. It, as I can imagine, acts like a low pass filter.
Yes. And, No.
I suppose that you are referring to very, very, very low order (with respect to the length of the dataset) polynomials. Due to the way polynomial fitting works, you quickly find yourself dealing with very small or very large numbers and numerical instabilities creep in. One second of data at 44.1KHz is 44100 data points. Fitting a 1024 order polynomial covers only about 2% of the original data points and would still require to raise numbers in the range $0 \ldots 1$ to the power of 1024.
Furthermore, please note that while you are shifting to the next window of interpolation (a new second of data comes in) you will have to now take into account constraints. That is, the beginning of the interpolation is not free to move anywhere it likes now. The curve must start as the last one ended to ensure continuity...Otherwise you get "clicks" as the waveform jumps at the transition. In fact, as far as polynomial fitting is concerned, you will most certainly get clicks because the curve will depart at whatever slope least squares dictates, in order to minimise the error and there, since you cannot know the future, it is very difficult to enforce constraints (that is, "no matter what, I want the curve to finish at a straight line segment with specific slope").
Fitting a parabola through it can be approximated as a half of a sinusoidal curve
No.
Where is the rest of the sinusoid and what will you do with a parabola "in between"? That is, as the peak slides off the window and we now enter the trough. A parabola could fit part of a sinusoid but it cannot substitute it.
This curve-fitting acts as a low pass filter. What would be the characteristics of this low pass filter?
To ask this, is to ask, how can I express one in terms of the other? That is, can I find an equivalence between polynomial fitting and low pass filtering as a reduced sum of trigonometric functions? And the answer is no because of the vastly different way that these two are structured.
Another thing to consider is the way that least square works, because, least squares will just strive to fit a parabola to the data even if it is not there. Well, what if my $n,x(n)$ doesn't bend it like a parabola? In fact, if your signal does not contain a component that varies with some combination of polynomial functions, the fit will fail. A prime example of this are impulsive functions. Take a beat that contains a bass drum (low frequencies) and a high hat (high frequencies) and try to "cut" the high hat using polynomial fitting. It's impossible. Polynomial fitting will try to make sense of everything, including the silence between the beats. The polynomial MUST fit. Least squares MUST find a local minimum.
An exception here might appear to be piecewise spline interpolation that split the waveform into parts and fit smaller polynomials, with constraints, between them but again, their definition doesn't allow for an easy transition between their spline representation and the Fourier Transform via which you could then "jump" to an impulse response. That is, to say that given the coefficients of a fitted piecewise spline, you can find a way to derive the impulse response of a Finite Impulse Response filter (let alone an Infinite Impulse Response filter).
You can always try to obtain a large sample of your data, fit a smoothing spline and then obtain the Fourier Transform of that to see what sort of low pass filter could approximate the result of the spline but that's not a way of deriving equivalent filters.
Hope this helps.