Given a particle moving uniformly on a circular path $\vec r(t)$, without loss of generality, we can parametrize the motion as:
$$ r(t) = R \big( \vec e_x \cos(\omega t) + \vec e_y \sin(\omega t) \big), $$
where $\omega$ is the angular velocity of the motion.
By using Newton's axiom $\vec F = m \vec a$ we can calculate the force necessary for the particle to move on that path (without even considering what might cause that force):
$$ \vec F(t) = m \partial_t^2 r(t) = - mR\omega^2 \big(\vec e_x \cos(\omega t) + \vec e_y \sin(\omega t) \big). $$
As one can easily see this force is directed towards the centre of the circle and has magnitude $m R \omega^2 = \frac{mv^2}{R}$. This proves that absolutely generally, for an object to move uniformly on a circular path there must be a centripetal force.
We can even generalize this argument further to a general path $\vec r(t)$. The force will still be given $\vec F(t) = m \ddot{\vec r}(t)$. Using a time dependent basis of the form $\vec e_t = \frac{\dot{\vec r}}{\left|\dot{\vec r}\right|}$, $\vec e_n = \frac{\ddot{\vec r} - (\ddot{\vec r} \cdot \vec e_t) \vec e_t}{\left|\ddot{\vec r} - (\ddot{\vec r} \cdot \vec e_t) \vec e_t\right|}$ and $\vec e_b = \vec e_t \times \vec e_n$ (the tangential, normal and binormal vectors) we can write:
$$ \vec F = \vec e_t F_t + \vec e_n F_n. $$
(No term with $\vec e_b$ appears as $\vec F \propto \ddot{\vec r}$ and $\ddot{\vec r} \in \text{span}\{\vec e_t, \vec e_n \}$ by construction).
Propositon $m\partial_t v = F_t$ and $F_n = \frac{mv^2}{R}$ (Notation: $\vec v = \dot{\vec r}$ and $v = \left| \vec v \right|$ as usual, $R$ is the radius of curvature which is independent of $v$).
Proof:
By direct calculation and using the above definitions, one easily gets:
$$m\partial_t \left| \dot{\vec r} \right| = m\partial_t \sqrt{\dot{\vec r} \cdot \dot{\vec r}} = m\frac{\ddot{\vec r} \cdot \dot{\vec r}}{\sqrt{\dot{\vec r} \cdot \dot{\vec r}}} = \vec F \cdot \vec e_t = F_t.$$
First we simplify the term for $\vec e_n$ using the identity $\vec a \times (\vec b \times \vec c) = \vec b (\vec a \cdot \vec c) - \vec c (\vec a \cdot \vec b)$:
$$\vec e_n = \frac{\ddot{\vec r} - (\ddot{\vec r} \cdot \vec e_t) \vec e_t}{\left|\ddot{\vec r} - (\ddot{\vec r} \cdot \vec e_t) \vec e_t\right|} = \frac{\ddot{\vec r} - \ddot{\vec r} (\vec e_t \cdot \vec e_t) + \vec e_t \times (\ddot{\vec r} \times \vec e_t)}{\left| \cdots \right|} = \frac{\vec e_t \times (\ddot{\vec r} \times \vec e_t)}{\left|\ddot{\vec r} \times \vec e_t\right|}$$
(Note: $\left|\vec e_t \times (\vec e_t \times \ddot{\vec r})\right| = \left|\vec e_t \times \ddot{\vec r}\right|$ because $\vec e_t \perp \vec e_t \times \ddot{\vec r}$ and $\left|\vec e_t\right| = 1$).
With this we arrive at (using $\vec a \cdot (\vec b \times \vec c) = \vec c \cdot (\vec a \times \vec b)$ in the second step):
\begin{align*}
F_n &= m \ddot{\vec r} \cdot \frac{\vec e_t \times (\ddot{\vec r} \times \vec e_t)}{\left|\ddot{\vec r} \times \vec e_t\right|} = m \frac{(\ddot{\vec r} \times \vec e_t)^2}{\left|\ddot{\vec r} \times \vec e_t\right|} = mv^2 \left|\frac{\ddot{\vec r} \times \vec e_t}{v^2}\right| = mv^2 \left|\frac{\ddot{\vec r} \times \dot{\vec r}}{v^3}\right|.
\end{align*}
From these terms we identify the radius of curvature $R$ as:
$$ R = \frac{v^3}{\left|\dot{\vec r} \times \ddot{\vec r}\right|}. $$
The proof that $R$ is constant under parameter changes $\vec r'(t) = (\vec r \circ f)(t)$ (with an arbitrary function $f$, that is it is independent of the velocity and only dependent on the curve traced by the path, formally this independence means $R'(t) = R\big(f(t)\big)$) is left as an exercise for the reader.
q.e.d.
In conclusion, this says, that for any movement along a path at any instance we can view the motion as an accelerated circular motion along the osculating circle of the path and only the tangential forces change the speed, while normal forces acting as centripetal force only change the direction of the velocity. So for the motion along any path we get, that if the motion is non-linear we will have a centripetal force whose parameters relate to the osculating circle.
Side note: The analysis of the geometry of the path are closely related to the first steps into the Frenet theory of the differential geometry of curves.
