If all the eigenvalues of an operator are real, is the operator Hermitian?

We shall assume that the vector space $V$ (where the linear operator $A:V\to V$ acts) is a complex vector space (as opposed to a real vector space), and that $V$ is equipped with a sesquilinear form $\langle\cdot,\cdot \rangle:V\times V \to \mathbb{C}$. (We will ignore subtleties with unbounded operators, domains, selfadjoint extensions, etc., in this answer.)

Since OP says he already knows how to prove that a Hermitian operator $A$ has real eigenvalues, he is essentially asking

If all the eigenvalues $\lambda_i$ are real, is the operator $A$ Hermitian?

The answer is No, only if $A$ is diagonalizable in an orthonormal basis.

In other words, the eigenspaces $\ker(A-\lambda_i {\bf 1})\subseteq V $ should be mutually orthogonal and together span the whole vector space $V$.

A version of the Spectral Theorem says that $A$ is orthonormally diagonalizable iff $A$ is a normal operator.

It's well known that it isn't true. Here is a counterexample: $\begin{pmatrix}1&1\\0&1\end{pmatrix}$. This is not hermitian, but it has two real eigenvalues +1,+1. This example is not diagonalizable, so it isn't so interesting.

A diagonalizable example is easy to construct too, if the eigenvectors are not orthogonal to each other. Consider the matrix $\begin{pmatrix}100&3\\-2&234\end{pmatrix}$. This matrix has two real eigenvalues close to 100 and 234, since the small perturbation of the eigenvalue equation doesn't change the discriminant. But the matrix is not symmetric, so it is not Hermitian. In this case, you can define a different metric on the vector space, a different definition of orthogonal, that makes the matrix Hermitian. This is easy-- the matrix is diagonal in it's Eigenbasis, with real eigenvalues, if you declare that this basis is orthonormal, then the matrix becomes Hermitian.

If you have a diagonalizable matrix with real eigenvalues $E_i$, and the eigenvectors $V_i$ are orthogonal and form a complete set,

Then the matrix is given by

$$ E_i \bar{V}_i^j V_i^k $$

This reconstructs a Hermitian matrix from the list of orthogonal real eigenvalues. A proper statement is that a diagonalizable matrix with real eigenvalues and a basis of eigenvectors defines a metric on the complex vector space where it becomes Hermitian. The proof is to declare that all the eigenvectors have zero inner product, and some positive norm.

In the subject of PT symmetric quantum mechanics, this construction defines the metric on Hilbert space from the energy eigenstates.