No – beginning in dimension 3 the two conditions are independent.
There are smooth manifolds which have

 • no non–trivial smooth (i.e. ) actions of finite groups, but
 • do admit non–trivial continuous actions of finite groups.

Consequently the absence of smooth actions does ​not​ imply the absence of continuous actions.

1. Why the implication holds in low dimensions

1–dimensional manifolds
A finite group acting continuously on or on a disjoint union of intervals is topologically conjugate to a group of smooth maps (classical results of Kerékjártó, Eilenberg).
Hence in dimension 1 “no smooth action” ⇔ “no continuous action”.

2–dimensional manifolds
For closed surfaces, every finite group of homeomorphisms is topologically conjugate to a group of diffeomorphisms (Kerékjártó, Brouwer, Nielsen).
Thus the implication also holds in dimension 2.

2. A wild finite-group action inside a ball

In dimension ≥3 there exist continuous finite–order homeomorphisms which are not even locally linear, let alone smooth. The prototype is Bing’s involution

 Bing (1952): an involution whose fixed-point set is a wildly  embedded Cantor set. A small modification of Bing’s construction gives an  involution

  σ : of order 2
  σ|∂B³ = id and Fix σ is a wild Cantor set contained in the interior.

(For higher dimensions take σ×id on .)

Because σ is the identity on the boundary, it can be inserted into any –manifold: if is a smoothly embedded closed ball, define

 τ(x) = { σ(x)  for x∈B^{n}
    x   for x∉B^{n}

Then τ is a continuous involution of whose fixed set contains the wild Cantor set; in particular the action is effective and cannot be topologically conjugate to a smooth action (a smooth or even locally linear involution would have a locally flat fixed-point set).

3. A manifold with no smooth finite-group actions

There are many closed smooth manifolds with no non-trivial smooth actions of finite groups – so-called asymmetric manifolds. Examples in every dimension ≥3 were constructed by

 R. Oliver, “Smooth manifolds with no effective smooth actions of a non-trivial finite group”, Proc. Amer. Math. Soc. 74 (1979), 136-138,

and later by Kwasik–Schultz, Puppe–Weinberger, etc. Call one such manifold (n≥3).

By definition, Diff contains no elements of finite order other than the identity, so admits no smooth finite-group actions.

4. Putting the two pieces together

Embed a small smooth -ball into and implant the involution σ described in §2. Extending by the identity outside we obtain a continuous, effective -action

 τ : .

Because its fixed set is wild, τ is not topologically conjugate to a smooth (or even locally linear) action. Yet still has no smooth finite-group actions at all.

Hence:

 A smooth manifold can have no smooth finite-group actions but still possess continuous ones.

5. Summary

• Dimensions 1 and 2: “No smooth finite-group action” ⇒ “No continuous finite-group action”.
• Dimension ≥ 3: The implication fails; the construction above gives counter-examples.

Therefore, for smooth manifolds of dimension at least three it is not sufficient to rule out smooth finite-group actions in order to rule out continuous ones.