Let $E \neq 0$ be a spectrum. Here is a classification of the strong $E$-local equivalences, a proof that the localization with respect to them exists, and an analysis of some special cases.
Classification of strong $E$-local equivalences:
Claim 1: A map of spaces $f: X \to Y$ is a strong $E$-local equivalence if and only if it is a $\pi_0$-equivalence and each map of connected components is a strong $E$-local equivalence.
The proof is straightfoward.
Claim 2: If $X,Y$ are connected spaces, then $f: X \to Y$ is a strong $E$-local equivalence if and only if $E_\ast(\Omega X) \to E_\ast(\Omega Y)$ is an isomorphism.
Proof: Note that $f^\ast: Sp_E^Y \to Sp_E^X$ is conservative and has a left adjoint $f_!$, so is an equivalence if and only if $M \to f^\ast f_! M$ is an equivalence for all $M \in Sp_E^Y$. Note that $Sp_E^X$ is equivalent to the category of modules for $L_E \Sigma^\infty_+ \Omega X \in Alg^{E_1}(Sp_E)$, and likewise $Sp_E^Y$ is the category of modules for $L_E \Sigma^\infty_+ \Omega Y$. So $f_!(M) = L_E \Sigma^\infty_+ \Omega Y \wedge_{L_E \Sigma^\infty_+ \Omega X} M$. So if $L_E \Sigma^\infty_+ \Omega X \to L_E \Sigma^\infty_+ \Omega Y$ is an equivalence, then $f^\ast$ is an equivalence, and taking $M = L_E \Sigma^\infty_+ \Omega X$, we see that the converse also holds. And this can be tested by taking $E$-homology, yielding the claim.
The local objects with respect to strong $E$-local equivalences form an accessible localization of all spaces:
Claim 3: Strong $E$-local equivalences are closed under 2-out-of-3 and under filtered colimits and coproducts in the arrow category.
The proof is straightforward.
Claim 4: Strong $E$-local equivalences are closed under cobase change (i.e. pushout along an arbitrary map).
For reference, consider a pushout square and its image under $\Omega$:
$$\require{AMScd}
(\ast) \begin{CD}
X @>>> Y\\
@VVV @VVV\\
W @>>> Z
\end{CD}
\qquad
(\ast\ast) \begin{CD}
\Omega X @>>> \Omega Y\\
@VVV @VVV\\
\Omega W @>>> \Omega Z
\end{CD}$$
Observation 5: The monad for $E_1$-spaces commutes with $\pi_0$, and preserves $E_\ast$-equivalences.
As a result,
Lemma 6: If $(\ast)$ is a pushout, all spaces are connected, and $X \to Y$ is a $\pi_1$-isomorphism, then $(\ast\ast)$ is also a pushout of $E_1$-spaces.
Proof:
By the equivalence of pointed connected spaces and grouplike $E_1$-spaces, it suffices to check that the pushout of the relevant $E_1$ spaces is grouplike. This can be checked on $\pi_0$, and so follows from the observation.
Lemma 7:
$E_\ast$-equivalences of $E_1$-spaces are stable under pushout.
Proof: Because the $E_1$ monad commutes with $E_\ast$-equivalences, it descends to the Bousfield localization at $E_\ast$-equivalences, and the result follows.
Proof of Claim 4: Suppose that $X \to Y$ is a strong $E$-local equivalence in $(\ast)$; we want to show that $W \to Z$ is as well. It suffices to consider the case where $W$ is connected. We may take the pushout in two stages: first we take the pushout along $\amalg_i X_i \to \vee_i X_i$ where the $X_i,Y_i$ are the connected components of $X,Y$. Second, we take the pushout along $\vee_i X_i \to W$, i.e. we consider the case where $X$ is connected.
The first case reduces to showing that if $X \to Y$ is a strong $E$-local equivalence of connected spaces, then so is $X \vee U \to Y \vee U$ where $U$ is connected, which reduces to the second case. Thus we may assume that $X$ is connected. But then by the two lemmas, because $\Omega X \to \Omega Y$ is an $E_\ast$-equivalence, so is $\Omega W \to \Omega Z$ as desired.
Claim 8: Every strong $E$-local equivalence is a highly filtered colimit of a small set of strong $E$-local equivalences.
Proof: This is clear in the category of pointed connected spaces, since $E_\ast$ and $\Omega$ are accessible functors. Then an arbitrary strong $E$-local equivalence is a coproduct strong $E$-local equivalences of pointed connected spaces, and because filtered colimits are computes the same in both categories the result follows.
Thus the strong $E$-local equivalences are an accessible subcategory of the arrow category of spaces closed under 2-out-of-3, colimits, and cobase change. It follows that the objects local with respect to them form an accessible localization of the category of spaces.
Some special cases:
Case A: Suppose that $f: X \to Y$ is a map of simply-connected spaces and an $H_\ast(-;R)$-equivalence, where $R$ is a PID. Then $\Omega f: \Omega X \to \Omega Y$ is also an $H_\ast(-;R)$-equivalence, i.e. $f$ is a strong $HR$-equivalence.
Proof: We have a map of Serre spectral sequences:
$$ H_p(X;R) \otimes_R H_q(\Omega X; R) \oplus Tor_1^R(H_{p-1}(X;R),H_q(\Omega X; R)) \Rightarrow R \\
\qquad \qquad \qquad \qquad \qquad\qquad \downarrow \qquad \qquad \qquad \qquad \qquad\qquad \qquad \quad = \\
H_p(Y;R) \otimes_R H_q(\Omega Y; R) \oplus Tor_1^R(H_{p-1}(Y;R),H_q(\Omega Y; R)) \Rightarrow R$$
Let $q$ be minimal such that $(\Omega f)_q: H_q(\Omega X;R) \to H_q(\Omega Y; R)$ is not an isomorphism. Then $q \geq 1$ because $\Omega X, \Omega Y$ are connected. If $(\Omega f)_q(\xi) = 0$ with $\xi\in H_q(\Omega X; R)$, then there are no differentials that can kill $1 \otimes \xi$, a contradiction. If $\zeta \in H_q(\Omega Y;R)$ is not in the image of $(\Omega f)_q$, then there are no differentials which can kill $1 \otimes \zeta$, another contradiction. So $(\Omega f)_\ast$ is an isomorphism as desired.
Case B: Let $\Phi$ be the Bousfield-Kuhn functor (for a fixed prime $p$ and $n\geq 1$). Suppose that $f: X \to Y$ is a map of connected spaces and $\Phi f$ is an equivalence. Then $f$ is a strong $T(n)$-equivalence (a similar statement holds for $L_{K(n)} \Phi$ and strong $K(n)$-equivalences).
Proof: $\Phi$ commutes with fiber sequences, so $\Phi(\Omega f)$ is an equivalence, so $T(n)_\ast \Omega f$ is an equivalence as desired.
